Advanced Math Alıştırma Soruları — SAT — Sayfa 10

Soru 181Soru →

In the $xy$ -plane, a system of equations consists of the circle with equation $(x - 7)^2 + (y - 4)^2 = 1$ and the line $y = kx$ , where $k$ is a positive constant. If the system has exactly one real solution and $k > 0.5$ , what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 0.75

Cevap

The correct answer is 3/4 (or 0.75).

The correct answer is

3/4

(or

0.75

). Substituting the line

y = kx

into the circle equation and setting the discriminant of the resulting quadratic equation to zero yields a quadratic in

k

:

48k^2 - 56k + 15 = 0

. Solving this equation gives two positive tangent slopes:

k = 5/12

and

k = 3/4

. Since the problem specifies that

k > 0.5

, we choose

k = 3/4

, which equals

0.75

. Alternatively, using geometry, the distance from the center

(7,4)

to the line

kx - y = 0

must equal the radius

1

. This gives

\frac{|7k-4|}{\sqrt{k^2+1}} = 1

. Squaring both sides yields

(7k-4)^2 = k^2+1

, which simplifies to

48k^2 - 56k + 15 = 0

, giving the same values of

k

.

Adım Adım Çözüm

1

Substitute the linear equation

y = kx

into the circle's quadratic equation

(x - 7)^2 + (y - 4)^2 = 1

.

(x - 7)^2 + (kx - 4)^2 = 1

To find the points of intersection between the line and the circle, substitute the expression for

y

from the linear equation into the circle's equation.

2

Expand the terms and write the equation in the standard quadratic form

Ax^2 + Bx + C = 0

.

(1 + k^2)x^2 - (14 + 8k)x + 64 = 0

Expanding

(x - 7)^2

yields

x^2 - 14x + 49

and expanding

(kx - 4)^2

yields

k^2x^2 - 8kx + 16

. Grouping the terms by powers of

x

and subtracting

1

from both sides gives the standard form.

3

Set the discriminant of the quadratic equation to

0

to satisfy the condition of exactly one real solution.

\Delta = [-(14 + 8k)]^2 - 4(1 + k^2)(64) = 0

A quadratic equation has exactly one real solution if and only if its discriminant,

B^2 - 4AC

, is equal to zero. Geometrically, this means the line is tangent to the circle.

4

Simplify the discriminant equation and solve for

k

.

48k^2 - 56k + 15 = 0

, which factors as

(12k - 5)(4k - 3) = 0

. Thus,

k = \frac{5}{12}

or

k = \frac{3}{4}

.

Simplifying

(14 + 8k)^2 - 256(1 + k^2) = 0

gives

196 + 224k + 64k^2 - 256 - 256k^2 = 0

, which simplifies to

-192k^2 + 224k - 60 = 0

. Dividing the entire equation by

-4

yields the quadratic equation in

k

.

5

Select the value of

k

that satisfies the given condition

k > 0.5

.

k = \frac{3}{4}

(or

0.75

)

Since

\frac{5}{12} \approx 0.417

is less than

0.5

and

\frac{3}{4} = 0.75

is greater than

0.5

, the correct value is

3/4

.

Anahtar Kavram

Solving a nonlinear system of equations involving a circle and a line by setting the discriminant of the substituted quadratic equation to zero to find the slope of the tangent lines.

Tahmini Süre:2m 30s

Soru 182Soru →

A circle and a line intersect at two points in the $xy$ -plane. The equations of the circle and the line are:

(x - 3)^2 + y^2 = 25

y = x - 2

If $(x_1, y_1)$ is the intersection point in Quadrant I and $(x_2, y_2)$ is the intersection point in Quadrant III, what is the value of $x_1 + y_2$ ?

3

5

1

10

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The value of

x_1 + y_2

is 3.

To solve the system, we substitute the linear equation

y = x - 2

into the circle equation, obtaining

(x - 3)^2 + (x - 2)^2 = 25

. Expanding this gives

x^2 - 6x + 9 + x^2 - 4x + 4 = 25

, which simplifies to

2x^2 - 10x - 12 = 0

. Dividing by 2, we get

x^2 - 5x - 6 = 0

, which factors as

(x - 6)(x + 1) = 0

. The solutions are

x = 6

and

x = -1

. Substituting these back into the linear equation gives the points of intersection:

(6, 4)

in Quadrant I (since both coordinates are positive) and

(-1, -3)

in Quadrant III (since both coordinates are negative). Therefore,

x_1 = 6

and

y_2 = -3

, and their sum is

6 + (-3) = 3

. This matches the correct option.

Adım Adım Çözüm

1

Substitute the linear equation into the circle equation to obtain an equation in terms of

x

only.

(x - 3)^2 + (x - 2)^2 = 25

This reduces the system to a single quadratic equation in one variable, allowing us to find the x-coordinates of the intersection points.

2

Expand and simplify the quadratic equation into standard form.

x^2 - 6x + 9 + x^2 - 4x + 4 = 25 \implies 2x^2 - 10x - 12 = 0 \implies x^2 - 5x - 6 = 0

Simplifying the equation makes it solvable by standard quadratic factoring methods.

3

Solve the simplified quadratic equation for

x

by factoring.

(x - 6)(x + 1) = 0 \implies x = 6

or

x = -1

The roots of this quadratic equation are the x-coordinates of the two intersection points.

4

Find the corresponding

y

-coordinates by substituting the

x

-values back into the linear equation

y = x - 2

.

For

x = 6

,

y = 6 - 2 = 4

, giving the point

(6, 4)

in Quadrant I. For

x = -1

,

y = -1 - 2 = -3

, giving the point

(-1, -3)

in Quadrant III.

This yields the coordinates of the two points of intersection and identifies which point lies in which quadrant.

5

Identify

x_1

and

y_2

and calculate the sum

x_1 + y_2

.

x_1 = 6

and

y_2 = -3

, so

x_1 + y_2 = 6 + (-3) = 3

.

This answers the specific question using the coordinate components of the two points.

Anahtar Kavram

Nonlinear Systems of Equations

Soru 183Soru →

If $9^{y - 1} \cdot 3^{2y} = 27^{y + 2}$ , what is the value of $y$ ?

1.5

4

7

8

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Cevap: 8

Cevap

8

To solve the equation, we rewrite each term using the common base of

3

. Since

9 = 3^2

and

27 = 3^3

, the equation

9^{y - 1} \cdot 3^{2y} = 27^{y + 2}

can be written as

(3^2)^{y-1} \cdot 3^{2y} = (3^3)^{y+2}

. Applying the power of a power rule, we get

3^{2(y-1)} \cdot 3^{2y} = 3^{3(y+2)}

, which simplifies to

3^{2y-2} \cdot 3^{2y} = 3^{3y+6}

. Using the product rule of exponents to combine the left side gives

3^{2y-2+2y} = 3^{4y-2} = 3^{3y+6}

. Setting the exponents equal to each other gives the linear equation

4y - 2 = 3y + 6

. Solving for

y

yields

y = 8

.

Adım Adım Çözüm

1

Express all bases in the equation as powers of 3.

9^{y-1} = (3^2)^{y-1} = 3^{2y-2}

and

27^{y+2} = (3^3)^{y+2} = 3^{3y+6}

. The original equation becomes

3^{2y-2} \cdot 3^{2y} = 3^{3y+6}

.

Having a common base allows us to combine exponents using the properties of exponents.

2

Simplify the left side of the equation by adding the exponents.

3^{(2y-2) + 2y} = 3^{4y-2}

, so the equation is

3^{4y-2} = 3^{3y+6}

.

When multiplying exponential expressions with the same base, we add their exponents.

3

Set the exponents equal to each other and solve for

y

.

4y - 2 = 3y + 6 \implies y - 2 = 6 \implies y = 8

.

Since the bases are equal and positive (and not equal to 1), their exponents must be equal.

Anahtar Kavram

Solving exponential equations by expressing terms with a common base and applying exponent laws.

Tahmini Süre:1m 30s

Soru 184Soru →

If $\left(\frac{1}{8}\right)^{2x - 3} = 16^{x - 2}$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 1.7

Cevap

1.7 (or 17/10)

The correct answer is 1.7 (or 17/10). By writing

\frac{1}{8}

as

2^{-3}

and

16

as

2^4

, the equation becomes

(2^{-3})^{2x - 3} = (2^4)^{x - 2}

. Applying exponent rules yields

2^{-6x + 9} = 2^{4x - 8}

. Equating the exponents gives

-6x + 9 = 4x - 8

, which simplifies to

10x = 17

, resulting in

x = 1.7

.

Adım Adım Çözüm

1

Express both bases as powers of

2

.

(2^{-3})^{2x - 3} = (2^4)^{x - 2}

To solve the exponential equation, both sides must have a common base. Both

\frac{1}{8}

and

16

can be rewritten as powers of

2

.

2

Apply the power-of-a-power exponent rule

(a^m)^n = a^{mn}

to simplify the exponents.

2^{-3(2x - 3)} = 2^{4(x - 2)}

, which simplifies to

2^{-6x + 9} = 2^{4x - 8}

This allows us to write each side as a single exponential term with a base of

2

.

3

Set the exponents equal to each other.

-6x + 9 = 4x - 8

Since the bases are equal, their exponents must be equal for the equation to hold.

4

Solve the linear equation for

x

.

10x = 17

, which gives

x = 1.7

Isolate the variable

x

by algebraic manipulation.

Anahtar Kavram

Solving exponential equations by expressing terms with a common base and applying exponent rules.

Alternatif Yöntem

Alternatively, you can apply the definition of logarithms to both sides of the equation. Taking the logarithm base 2 of both sides gives

\log_2\left(\left(\frac{1}{8}\right)^{2x-3}\right) = \log_2\left(16^{x-2}\right)

. Using the power property of logarithms, this simplifies to

(2x - 3)\log_2\left(2^{-3}\right) = (x - 2)\log_2\left(2^4\right)

, which simplifies directly to

-3(2x - 3) = 4(x - 2)

. Solving this linear equation yields the same result,

x = 1.7

.

Tahmini Süre:1m 30s

Soru 185Soru →

The profit $P(x)$ , in dollars, a company makes from selling $x$ units of a product is given by the function $P(x) = -3x^2 + bx - 1,500$ , where $b$ is a positive constant. If the maximum profit the company can make is $1,200$ dollars, what is the value of $b$ ?

Cevabı ve açıklamayı göster

Cevap: 180

Cevap

180

The maximum profit of the quadratic profit function occurs at its vertex. The x-coordinate of the vertex of

P(x) = -3x^2 + bx - 1,500

is given by

x = -\frac{b}{2(-3)} = \frac{b}{6}

. Substituting this into the profit equation and setting it equal to

1,200

yields

-3\left(\frac{b}{6}\right)^2 + b\left(\frac{b}{6}\right) - 1,500 = 1,200

. Simplifying this expression results in

\frac{b^2}{12} = 2,700

, which gives

b^2 = 32,400

. Since

b

is positive,

b = 180

.

Adım Adım Çözüm

1

Identify the x-coordinate of the vertex of the quadratic function in terms of

b

.

The x-coordinate of the vertex is

x = \frac{b}{6}

.

A quadratic function in the form

f(x) = ax^2 + bx + c

has its maximum value at the vertex when the coefficient of the squared term,

a

, is negative. The x-coordinate of the vertex is given by

x = -\frac{b}{2a}

. For this function,

a = -3

, so the x-coordinate of the vertex is

x = -\frac{b}{2(-3)} = \frac{b}{6}

.

2

Substitute

x = \frac{b}{6}

back into the profit equation

P(x)

and set it equal to the maximum profit of

1,200

dollars.

-3\left(\frac{b}{6}\right)^2 + b\left(\frac{b}{6}\right) - 1,500 = 1,200

The maximum profit of

1,200

dollars is the value of the function at the vertex.

3

Simplify the equation to solve for the positive constant

b

.

b = 180

Simplifying the term

-3\left(\frac{b}{6}\right)^2

gives

-3\left(\frac{b^2}{36}\right) = -\frac{b^2}{12}

. Simplifying the term

b\left(\frac{b}{6}\right)

gives

\frac{b^2}{6}

. Substituting these back in gives

-\frac{b^2}{12} + \frac{b^2}{6} - 1,500 = 1,200

. Combining the

b^2

terms gives

\frac{b^2}{12} - 1,500 = 1,200

. Adding

1,500

to both sides gives

\frac{b^2}{12} = 2,700

. Multiplying both sides by

12

yields

b^2 = 32,400

. Taking the square root of both sides gives

b = \pm 180

. Since

b

must be a positive constant,

b = 180

.

Anahtar Kavram

Finding the maximum value of a quadratic function by using the vertex formula

x = -\frac{b}{2a}

and evaluating the function at that point.

Soru 186Soru →

For all $x > 3$ , the expression $\frac{2x^2 - 11x + 15}{x - 3} + \frac{3x^2 - 12}{x - 2}$ is equivalent to $ax + b$ , where $a$ and $b$ are constants. What is the value of $a + b$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The value of

a + b

is

6

.

The correct answer is

6

. By factoring the numerators of both rational expressions, we can simplify them. The first term is

\frac{(2x-5)(x-3)}{x-3} = 2x-5

. The second term is

\frac{3(x-2)(x+2)}{x-2} = 3(x+2) = 3x+6

. Adding these simplified terms gives

(2x-5) + (3x+6) = 5x+1

. Comparing this to

ax+b

yields

a=5

and

b=1

, so

a+b = 5+1=6

.

Adım Adım Çözüm

1

Factor the numerator of the first rational expression and simplify.

\frac{2x^2 - 11x + 15}{x - 3} = 2x - 5

Since

2x^2 - 11x + 15 = (2x - 5)(x - 3)

, the factor

(x - 3)

divides out for all

x > 3

.

2

Factor the numerator of the second rational expression and simplify.

\frac{3x^2 - 12}{x - 2} = 3x + 6

Since

3x^2 - 12 = 3(x - 2)(x + 2)

, the factor

(x - 2)

divides out for all

x > 3

.

3

Combine the simplified terms by adding them.

5x + 1

Adding

(2x - 5)

and

(3x + 6)

yields

(2x + 3x) + (-5 + 6) = 5x + 1

.

4

Identify the values of

a

and

b

and calculate their sum.

a = 5

,

b = 1

, and

a + b = 6

Comparing

5x + 1

with

ax + b

gives

a = 5

and

b = 1

, so

a + b = 5 + 1 = 6

.

Anahtar Kavram

Simplifying rational expressions by factoring the numerator and dividing out common factors.

Soru 187Soru →

For what value of $x$ is the equation $8^{x + 1} = 16^x$ true?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The correct answer is 3.

The correct answer is 3. To find the value of

x

, express both 8 and 16 as powers of 2:

8 = 2^3

and

16 = 2^4

. Substituting these values into the equation gives

(2^3)^{x + 1} = (2^4)^x

. Applying the power of a power rule,

(a^m)^n = a^{mn}

, yields

2^{3(x + 1)} = 2^{4x}

, which simplifies to

2^{3x + 3} = 2^{4x}

. Since the bases are now the same, their exponents must be equal:

3x + 3 = 4x

. Subtracting

3x

from both sides gives

x = 3

.

Adım Adım Çözüm

1

Express the bases 8 and 16 as powers of 2.

(2^3)^{x + 1} = (2^4)^x

Writing both sides of the equation with a common base allows for the equating of exponents.

2

Apply the power of a power rule

(a^m)^n = a^{mn}

to simplify the exponents.

2^{3x + 3} = 2^{4x}

This simplifies each side of the equation to a single base with a single exponent.

3

Equate the exponents and solve the resulting linear equation.

3x + 3 = 4x

, which simplifies to

x = 3

Since the bases are equal, their exponents must be equal for the equation to hold true.

Anahtar Kavram

Solving exponential equations by expressing both sides with a common base and equating the exponents.

Alternatif Yöntem

An alternative approach is to write the equation in terms of base 4. While 8 is not an integer power of 4, we can write

8 = 4^{1.5}

and

16 = 4^2

. The equation becomes

(4^{1.5})^{x+1} = (4^2)^x

, which simplifies to

1.5(x+1) = 2x

. Solving this gives

1.5x + 1.5 = 2x \implies 0.5x = 1.5 \implies x = 3

.

Tahmini Süre:45s

Soru 188Soru →

What is the set of all solutions to the equation $\sqrt{x + 12} = x$ ?

4

only

-3

only

4

and

-3

3

and

-4

Cevabı ve açıklamayı göster

Cevap: $4$ only

Cevap

The solution set containing only

4

The correct answer is the set containing only

4

. Squaring both sides of the equation

\sqrt{x + 12} = x

produces

x + 12 = x^2

. Rearranging this into standard quadratic form gives

x^2 - x - 12 = 0

. Factoring the quadratic gives

(x - 4)(x + 3) = 0

, which yields potential solutions of

x = 4

and

x = -3

. Testing

x = 4

in the original equation yields

\sqrt{4 + 12} = \sqrt{16} = 4

, which is a true statement. Testing

x = -3

yields

\sqrt{-3 + 12} = \sqrt{9} = 3

, which does not equal

-3

. Thus,

x = -3

is extraneous, and

4

is the only valid solution.

Adım Adım Çözüm

1

Square both sides of the equation to eliminate the radical.

x + 12 = x^2

Squaring both sides of

\sqrt{x + 12} = x

removes the square root.

2

Rewrite the equation in standard quadratic form

ax^2 + bx + c = 0

.

x^2 - x - 12 = 0

Subtracting

x

and

12

from both sides moves all terms to one side of the equation.

3

Factor the quadratic equation.

(x - 4)(x + 3) = 0

We search for two numbers that multiply to

-12

and add to

-1

. These numbers are

-4

and

3

.

4

Solve for the potential solutions.

x = 4

or

x = -3

Setting each factor equal to zero yields the prospective values of

x

.

5

Check both potential solutions in the original equation to identify extraneous solutions.

Substituting

x = 4

:

\sqrt{4 + 12} = \sqrt{16} = 4

(valid). Substituting

x = -3

:

\sqrt{-3 + 12} = \sqrt{9} = 3 \neq -3

(extraneous).

Squaring both sides of an equation can introduce extraneous roots that must be discarded.

Anahtar Kavram

Solving radical equations and verifying solutions to eliminate extraneous roots

Soru 189Soru →

The graph of the quadratic function $f$ in the $xy$ -plane has a vertex at $(4, 7)$ . If the function $g$ is defined by $g(x) = f(x) + 5$ , what is the $y$ -coordinate of the vertex of the graph of $y = g(x)$ ?

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

The correct answer is 12.

The vertex of the graph of the quadratic function

f

is

(4, 7)

. The function

g(x) = f(x) + 5

adds

5

to every output value of

f(x)

, which shifts the entire graph vertically upward by

5

units. Consequently, the vertex of the graph of

g

is translated from

(4, 7)

to

(4, 7 + 5)

, which simplifies to

(4, 12)

. Therefore, the

y

-coordinate of the vertex of the graph of

y = g(x)

is

12

.

Adım Adım Çözüm

1

Identify the y-coordinate of the vertex of the graph of the function f.

The y-coordinate of the vertex of the graph of f is 7.

The vertex of the graph of f is given as (4, 7).

2

Determine the vertical translation from f(x) to g(x).

The graph of g(x) is shifted upward by 5 units relative to the graph of f(x).

The term + 5 in the definition g(x) = f(x) + 5 increases all y-values of the function by 5.

3

Calculate the y-coordinate of the vertex of the graph of g.

The new y-coordinate is 7 + 5 = 12.

Applying the vertical shift of 5 units to the original y-coordinate of 7 gives the new vertex's y-coordinate.

Anahtar Kavram

Vertical translations of quadratic functions and their graphs

Soru 190Soru →

The quadratic function $h$ is defined by $h(x) = (x + 2)^2 - 5$ . What are the coordinates of the vertex of the graph of $y = h(x)$ in the $xy$ -plane?

(-2, -5)

(2, -5)

(-2, 5)

(2, 5)

Cevabı ve açıklamayı göster

Cevap: $(-2, -5)$

Cevap

The coordinates of the vertex of the graph are

(-2, -5)

.

The correct answer is the coordinate pair

(-2, -5)

. The vertex form of a quadratic function is given by

y = a(x - h)^2 + k

, where

(h, k)

is the vertex of the parabola. For the given equation

h(x) = (x + 2)^2 - 5

, we rewrite the squared expression as

(x - (-2))^2

to match the standard form. This shows that

h = -2

. The value of

k

is the constant term

-5

. Therefore, the vertex of the graph of

h

is

(-2, -5)

.

Adım Adım Çözüm

1

Identify the general vertex form of a quadratic function.

The vertex form is

y = a(x - h)^2 + k

, where the vertex of the parabola is at the point

(h, k)

.

This general equation allows us to directly read the coordinates of the vertex by comparing the coefficients.

2

Compare the given function

h(x) = (x + 2)^2 - 5

to the vertex form.

By matching the terms, we get

x - h = x + 2

, which implies

h = -2

. The constant term is

k = -5

.

This identifies the values of

h

and

k

that make up the coordinates of the vertex

(h, k)

.

3

Write the vertex coordinates.

The vertex coordinates are

(h, k) = (-2, -5)

.

Combining the identified values gives the final coordinates of the vertex.

Anahtar Kavram

Identifying the vertex of a quadratic function directly from its vertex form equation,

y = a(x - h)^2 + k

.

Soru 191Soru →

If a polynomial $p(x)$ is divided by $x - 5$ , the remainder is $12$ . Which of the following equations must be true?

p(5) = 12

p(-5) = 12

p(5) = -12

p(12) = 5

Cevabı ve açıklamayı göster

Cevap: $p(5) = 12$

Cevap

p(5) = 12

According to the Remainder Theorem, dividing a polynomial

p(x)

by a linear expression

x - c

results in a remainder equal to

p(c)

. Here, the divisor is

x - 5

, so

c = 5

. Since the remainder is

12

, the equation that must be true is

p(5) = 12

.

Adım Adım Çözüm

1

Identify the divisor and its corresponding value of

c

using the Remainder Theorem.

The divisor is

x - 5

, so we set

x - 5 = 0

, which gives

c = 5

.

The Remainder Theorem states that the remainder of a polynomial

p(x)

divided by

x - c

is

p(c)

.

2

Set

p(c)

equal to the given remainder.

Since the remainder is

12

and

c = 5

, we get

p(5) = 12

.

This directly applies the relation

p(c) = \text{remainder}

.

Anahtar Kavram

The Remainder Theorem

Soru 192Soru →

For the exponential function $g(x) = k \cdot a^x$ , where $k$ and $a$ are positive constants, the value of $g(x + 1)$ is $20\%$ greater than the value of $g(x - 1)$ for all real numbers $x$ . If $g(2) = 15$ , what is the value of $g(6)$ ?

21.0

21.6

31.104

36.0

Cevabı ve açıklamayı göster

Cevap: 21.6

Cevap

21.6

The correct value of 21.6 is obtained by setting up the ratio

g(x+1)/g(x-1) = a^2

. Since

g(x+1)

is

20\%

greater than

g(x-1)

, this ratio is equal to 1.20, so

a^2 = 1.20

. Since

g(6) = g(2) \cdot a^4 = 15 \cdot (a^2)^2

, substituting

a^2 = 1.20

gives

15 \cdot (1.20)^2 = 15 \cdot 1.44 = 21.6

.

Adım Adım Çözüm

1

Express the values of the function at x + 1 and x - 1 using the definition of g(x).

g(x + 1) = k \cdot a^{x + 1}

and

g(x - 1) = k \cdot a^{x - 1}

To set up the mathematical relationship between the two values based on the function definition.

2

Apply the condition that g(x + 1) is 20% greater than g(x - 1).

k \cdot a^{x + 1} = 1.20 \cdot k \cdot a^{x - 1}

, which simplifies to

a^2 = 1.20

To find the factor of growth over an interval of 2 units of x.

3

Relate the value of g(6) to the given value of g(2).

g(6) = g(2) \cdot a^4 = g(2) \cdot (a^2)^2

To write the unknown value in terms of the known value and the determined factor

a^2

using exponent rules.

4

Substitute the known values into the equation to calculate the result.

g(6) = 15 \cdot (1.20)^2 = 15 \cdot 1.44 = 21.6

To calculate the final numerical value of g(6).

Anahtar Kavram

Exponential Functions and Equations

Tahmini Süre:2m 30s

Soru 193Soru →

A parabola in the $xy$ -plane has a vertex at $(3, -4)$ and passes through the point $(1, 0)$ . A second parabola is created by reflecting the original parabola across the $x$ -axis, then translating the resulting graph $4$ units to the left and $2$ units down. If this second parabola represents the graph of the function $g$ , which of the following equations defines $g(x)$ ?

g(x) = -(x - 7)^2 + 2

g(x) = -(x + 1)^2 + 2

g(x) = -(x + 1)^2 - 6

g(x) = (x + 1)^2 - 6

Cevabı ve açıklamayı göster

Cevap: $g(x) = -(x + 1)^2 + 2$

Cevap

The equation

g(x) = -(x + 1)^2 + 2

The correct equation is

g(x) = -(x + 1)^2 + 2

. Since the vertex of the original parabola

f(x)

is

(3, -4)

and it passes through

(1, 0)

, its vertex form equation is

f(x) = (x - 3)^2 - 4

. Reflecting

f(x)

across the

x

-axis negates the entire function, resulting in

-(x - 3)^2 + 4

. Shifting this function

4

units to the left replaces

x

with

x + 4

, yielding

-(x + 1)^2 + 4

. Finally, shifting it

2

units down subtracts

2

from the entire expression, giving

g(x) = -(x + 1)^2 + 2

.

Adım Adım Çözüm

1

Determine the equation of the original parabola

f(x)

using its vertex form

f(x) = a(x - h)^2 + k

.

f(x) = (x - 3)^2 - 4

Since the vertex is

(3, -4)

, the vertex form is

f(x) = a(x - 3)^2 - 4

. Substituting the point

(1, 0)

gives

0 = a(1 - 3)^2 - 4 \implies 4a = 4 \implies a = 1

.

2

Reflect the graph of

f(x)

across the

x

-axis.

-f(x) = -(x - 3)^2 + 4

A reflection across the

x

-axis negates the entire function, so

y = -f(x)

.

3

Translate the reflected graph

4

units to the left.

-(x + 1)^2 + 4

Translating a function

h(x)

to the left by

c

units is represented by

h(x + c)

. Replacing

x

with

x + 4

in

-(x - 3)^2 + 4

gives

-((x + 4) - 3)^2 + 4 = -(x + 1)^2 + 4

.

4

Translate the graph

2

units down to find the final function

g(x)

.

g(x) = -(x + 1)^2 + 2

Translating a function down by

d

units is represented by subtracting

d

from the function. Subtracting

2

from

-(x + 1)^2 + 4

gives

-(x + 1)^2 + 2

.

Anahtar Kavram

Applying transformations (reflections and horizontal/vertical translations) to quadratic functions in vertex form.

Soru 194Soru →

The quadratic function $f$ is defined by $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, and has its vertex at $(2, 4)$ . The function $g$ is defined by $g(x) = f(x - 3) + k$ , where $k$ is a constant. The vertex of the graph of $y = g(x)$ in the $xy$ -plane lies on the line $y = 2x - 3$ . If the product of the $x$ -intercepts of the graph of $y = g(x)$ is $11$ , what is the $y$ -intercept of the graph of $y = f(x)$ ?

2

6

-4

-6

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

2

The correct answer is 2. By writing the quadratic function

f(x)

in vertex form as

f(x) = a(x - 2)^2 + 4

, we can apply the transformation rules to express

g(x) = f(x - 3) + k

as having a vertex at

(5, 4 + k)

. Since this vertex lies on the line

y = 2x - 3

, substituting

x = 5

gives

4 + k = 7

, meaning

g(x) = a(x - 5)^2 + 7

. The roots of

g(x) = 0

are

5 \pm \sqrt{-7/a}

, and their product is

25 + 7/a

. Setting this product equal to 11 yields

a = -1/2

. Substituting

a = -1/2

back into the expression for

f(0) = 4a + 4

gives 2.

Adım Adım Çözüm

1

Write the function

f(x)

in vertex form using its vertex

(2, 4)

.

f(x) = a(x - 2)^2 + 4

Any quadratic function with a vertex at

(h, k)

can be written in the form

f(x) = a(x - h)^2 + k

.

2

Determine the vertex of the transformed function

g(x) = f(x - 3) + k

.

The vertex of

g(x)

is

(5, 4 + k)

.

The transformation

f(x - 3)

shifts the graph 3 units to the right, changing the

x

-coordinate of the vertex from 2 to 5. The addition of

k

shifts the graph vertically by

k

units, changing the

y

-coordinate of the vertex from 4 to

4 + k

.

3

Use the condition that the vertex of

y = g(x)

lies on the line

y = 2x - 3

to find the value of

k

and the vertex coordinates.

4 + k = 7 \implies k = 3

. The vertex of

g(x)

is

(5, 7)

.

Since the vertex

(5, 4 + k)

lies on the line

y = 2x - 3

, substituting

x = 5

into the line equation gives

y = 2(5) - 3 = 7

.

4

Write

g(x)

in vertex form and express its roots (the

x

-intercepts) in terms of

a

.

g(x) = a(x - 5)^2 + 7

. The roots are

x = 5 \pm \sqrt{-\frac{7}{a}}

.

Setting

g(x) = 0

gives

a(x - 5)^2 + 7 = 0

, which simplifies to

(x - 5)^2 = -\frac{7}{a}

, yielding

x = 5 \pm \sqrt{-\frac{7}{a}}

.

5

Calculate the product of the roots and set it equal to 11 to solve for

a

.

25 + \frac{7}{a} = 11 \implies a = -\frac{1}{2}

.

The product of the roots is

(5 + \sqrt{-\frac{7}{a}})(5 - \sqrt{-\frac{7}{a}}) = 25 - (-\frac{7}{a}) = 25 + \frac{7}{a}

. Setting this equal to 11 gives

\frac{7}{a} = -14

, so

a = -\frac{1}{2}

.

6

Evaluate

f(0)

to find the

y

-intercept of the graph of

f(x)

.

f(0) = 4(-\frac{1}{2}) + 4 = 2

.

The

y

-intercept is the value of the function at

x = 0

. Substituting

x = 0

and

a = -\frac{1}{2}

into

f(x) = a(x - 2)^2 + 4

yields

4a + 4 = 2

.

Anahtar Kavram

Quadratic transformations and properties of roots in vertex form

Soru 195Soru →

\begin{aligned} y &= 2x^2 - 5x - 1 \\ y &= 3x - 7 \end{aligned}

If $(x, y)$ is a solution to the system of equations above and $y > 0$ , what is the value of $x + y$ ?

-11

5

-3

1

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5

Substituting

y = 3x - 7

into

y = 2x^2 - 5x - 1

yields the quadratic equation

3x - 7 = 2x^2 - 5x - 1

. Moving all terms to one side gives

2x^2 - 8x + 6 = 0

. Dividing by 2 simplifies this to

x^2 - 4x + 3 = 0

, which factors as

(x - 1)(x - 3) = 0

. This gives the solutions

x = 1

and

x = 3

. Substituting these into the linear equation yields the coordinate points

(1, -4)

and

(3, 2)

. Since the problem states that

y > 0

, we choose the solution

(3, 2)

. The value of

x + y

is

3 + 2 = 5

.

Adım Adım Çözüm

1

Substitute the expression for

y

from the linear equation into the quadratic equation.

3x - 7 = 2x^2 - 5x - 1

Since both equations are solved for

y

, they can be set equal to each other to find the

x

-coordinates of the intersection points.

2

Rearrange the terms to set the quadratic equation equal to zero, and then divide by the common factor.

2x^2 - 8x + 6 = 0

, which simplifies to

x^2 - 4x + 3 = 0

Putting the equation in standard form

ax^2 + bx + c = 0

allows us to factor and solve for

x

.

3

Factor the quadratic equation to find the values of

x

.

(x - 1)(x - 3) = 0

, so

x = 1

or

x = 3

Factoring allows us to find the roots of the equation directly.

4

Substitute the

x

values back into the linear equation to solve for the corresponding

y

values.

For

x = 1

,

y = 3(1) - 7 = -4

, giving the solution

(1, -4)

. For

x = 3

,

y = 3(3) - 7 = 2

, giving the solution

(3, 2)

.

Finding the

y

-coordinates completes the solutions

(x, y)

to the system.

5

Apply the given constraint

y > 0

and calculate

x + y

.

The solution

(3, 2)

satisfies

y > 0

since

2 > 0

. Thus,

x + y = 3 + 2 = 5

.

The problem asks for the sum of the coordinates of the solution that has a positive

y

-value.

Anahtar Kavram

Solving a system consisting of a linear equation and a quadratic equation by substitution and applying coordinate constraints.

Tahmini Süre:1m 30s

Soru 196Soru →

A quadratic function $f$ is defined by $f(x) = a(x - h)^2 + k$ , where $a$ , $h$ , and $k$ are constants. The graph of $y = f(x)$ in the $xy$ -plane passes through the points $(0, 5)$ and $(4, 5)$ . If the minimum value of $f(x)$ for $0 \le x \le 3$ is $1$ , what is the value of $f(6)$ ?

Cevabı ve açıklamayı göster

Cevap: 17

Cevap

The value of

f(6)

is

17

.

The correct answer is

17

. Since the quadratic function passes through

(0, 5)

and

(4, 5)

, its axis of symmetry is the line

x = 2

, which means the vertex

x

-coordinate is

h = 2

. For the interval

0 \le x \le 3

, this vertex is within the bounds. An upward-opening parabola has its minimum value at its vertex, so the minimum value of

1

must be the

y

-coordinate of the vertex, giving

k = 1

. The function can then be written as

f(x) = a(x - 2)^2 + 1

. Substituting

(0, 5)

into this equation gives

5 = a(0 - 2)^2 + 1

, which simplifies to

4a = 4

, meaning

a = 1

. The fully determined function is

f(x) = (x - 2)^2 + 1

. Evaluating this at

x = 6

yields

f(6) = (6 - 2)^2 + 1 = 17

.

Adım Adım Çözüm

1

Determine the axis of symmetry of the parabola.

h = 2

Since the points

(0, 5)

and

(4, 5)

have the same

y

-value, they are symmetric across the vertical line of symmetry of the parabola. The

x

-coordinate of the vertex

h

is the average of the

x

-coordinates of these two points:

h = \frac{0 + 4}{2} = 2

.

2

Determine the

y

-coordinate of the vertex

k

using the minimum value over the interval

[0, 3]

.

k = 1

The vertex

x

-coordinate

x = 2

lies within the interval

0 \le x \le 3

. For a quadratic function that opens upward, the minimum value occurs at the vertex. Thus, the minimum value on this interval is the vertex

y

-coordinate, so

k = 1

.

3

Substitute the point

(0, 5)

into the vertex form of the equation to solve for

a

.

a = 1

Using the vertex form

f(x) = a(x - 2)^2 + 1

, substitute

x = 0

and

f(x) = 5

:

5 = a(0 - 2)^2 + 1 \implies 5 = 4a + 1 \implies 4a = 4 \implies a = 1

.

4

Evaluate

f(6)

using the complete quadratic function formula.

f(6) = 17

Substitute

x = 6

into the equation

f(x) = (x - 2)^2 + 1

:

f(6) = (6 - 2)^2 + 1 = 4^2 + 1 = 16 + 1 = 17

.

Anahtar Kavram

Using symmetry properties and interval extrema to determine the equation of a quadratic function in vertex form.

Soru 197Soru →

\begin{aligned} y + 11 &= x^2 \\ y - 3x &= 7 \end{aligned}

If $(x, y)$ is a solution to the system of equations above and $x > 0$ , what is the value of $x + y$ ?

Cevabı ve açıklamayı göster

Cevap: 31

Cevap

The correct answer is 31.

Solving the system of equations by expressing

y

in terms of

x

from the linear equation gives

y = 3x + 7

. Substituting this expression for

y

in the first equation yields

(3x + 7) + 11 = x^2

, which simplifies to

x^2 - 3x - 18 = 0

. Factoring the quadratic expression gives

(x - 6)(x + 3) = 0

, so

x = 6

or

x = -3

. Since the problem specifies that

x > 0

, the

x

-value of the solution must be

6

. Substituting

x = 6

back into the linear equation gives

y = 3(6) + 7 = 25

. The sum of

x

and

y

is

6 + 25 = 31

.

Adım Adım Çözüm

1

Express

y

in terms of

x

from the linear equation.

y = 3x + 7

This isolates the variable

y

to facilitate substitution.

2

Substitute the expression for

y

into the quadratic equation.

(3x + 7) + 11 = x^2

This eliminates the variable

y

, yielding a single equation in terms of

x

.

3

Rearrange the equation into standard quadratic form.

x^2 - 3x - 18 = 0

Standard form (

ax^2 + bx + c = 0

) is required to solve by factoring.

4

Factor the quadratic equation.

(x - 6)(x + 3) = 0

Finding factors helps determine the possible values of

x

.

5

Find the values of

x

and apply the positive constraint.

x = 6

The equation yields

x = 6

and

x = -3

. The condition

x > 0

restricts the solution to

x = 6

.

6

Calculate the corresponding value of

y

.

y = 25

Substituting

x = 6

into the linear equation gives

y = 3(6) + 7 = 25

.

7

Calculate the sum of

x

and

y

.

31

The problem asks for the value of

x + y

.

Anahtar Kavram

Solving systems of nonlinear equations algebraically using substitution and quadratic factoring.

Alternatif Yöntem

Alternatively, solve the linear equation for

x

to get

x = \frac{y - 7}{3}

and substitute this into the quadratic equation to solve for

y

first. This approach is more complex because it introduces fractional terms.

Tahmini Süre:1m 30s

Soru 198Soru →

If $x$ and $y$ are positive numbers with $x > 1$ that satisfy the system of equations below, what is the value of $x$ ?

x^y = 2^{12}

(2x)^{y-2} = 2^8

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The correct answer is 8.

The correct answer is 8. By manipulating the system of equations, we can express the second equation in terms of

x^y

. Substituting the value

2^{12}

for

x^y

allows us to express

x

in terms of

y

as

x = 2^{(y+2)/2}

. Substituting this back into the first equation results in the quadratic equation

y^2 + 2y - 24 = 0

. Since

y

must be positive, we find

y = 4

. Using this value of

y

, we solve for

x

and obtain

x = 8

, which is greater than 1, satisfying all given conditions.

Adım Adım Çözüm

1

Distribute the exponent in the second equation:

(2x)^{y-2} = 2^8

.

2^{y-2} \cdot x^{y-2} = 2^8

Applying the power of a product rule:

(ab)^n = a^n b^n

.

2

Multiply both sides of the equation by

x^2

to align with the term

x^y

.

2^{y-2} \cdot x^y = 2^8 \cdot x^2

Since

x^{y-2} \cdot x^2 = x^{y-2+2} = x^y

, this allows us to substitute the value of

x^y

from the first equation.

3

Substitute

x^y = 2^{12}

into the equation.

2^{y-2} \cdot 2^{12} = 2^8 \cdot x^2

This uses the given system's first equation to eliminate the

x^y

term.

4

Simplify the left side and divide both sides by

2^8

to isolate

x^2

.

x^2 = 2^{y+2}

2^{y-2} \cdot 2^{12} = 2^{y+10}

. Dividing both sides by

2^8

yields

2^{y+10-8} = 2^{y+2}

.

5

Take the square root of both sides to find an expression for

x

.

x = 2^{\frac{y+2}{2}}

Since

x > 1

, we take the positive root:

x = (2^{y+2})^{1/2} = 2^{\frac{y+2}{2}}

.

6

Substitute the expression for

x

back into the first equation

x^y = 2^{12}

.

\left(2^{\frac{y+2}{2}}\right)^y = 2^{12}

This yields a single equation in terms of

y

.

7

Equate the exponents and solve the resulting quadratic equation for

y

.

y = 4

(since

y > 0

)

The equation is

2^{\frac{y^2+2y}{2}} = 2^{12} \implies \frac{y^2+2y}{2} = 12 \implies y^2+2y-24=0 \implies (y+6)(y-4)=0

. Since

y

must be positive,

y = 4

.

8

Substitute

y = 4

back into the expression for

x

.

x = 8

x = 2^{\frac{4+2}{2}} = 2^3 = 8

.

Anahtar Kavram

Solving systems of exponential equations by equating bases, applying exponent rules, and solving quadratic equations.

Soru 199Soru →

A biologist is studying a population of bacteria that triples in size every 4 hours. The population of the bacteria $t$ hours after the start of the study can be modeled by the function $P(t) = P_0 \cdot 3^{kt}$ , where $P_0$ is the initial population and $k$ is a constant. What is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 0.25

Cevap

The correct answer is 0.25 (or 1/4).

The population triples every 4 hours, meaning that after

t

hours, the population has tripled

\frac{t}{4}

times. Thus, the growth model is

P(t) = P_0 \cdot 3^{\frac{t}{4}}

. Comparing this to the given expression

P(t) = P_0 \cdot 3^{kt}

, we get

kt = \frac{t}{4}

. Dividing both sides by

t

yields

k = \frac{1}{4}

, or

0.25

.

Adım Adım Çözüm

1

Write the general formula for exponential growth where the population triples every 4 hours.

P(t) = P_0 \cdot 3^{\frac{t}{4}}

If a population triples every 4 hours, it will be multiplied by 3 a total of

\frac{t}{4}

times after

t

hours.

2

Set the exponent of the general model equal to the exponent of the given model.

kt = \frac{t}{4}

Both models represent the same population growth function, so their exponents must be equal.

3

Solve for the constant

k

.

k = \frac{1}{4}

(or

0.25

)

Divide both sides of the equation by

t

.

Anahtar Kavram

Exponential growth functions and representing time intervals in the exponent.

Soru 200Soru →

If $16^{x-1} - 10 \cdot 4^{x-2} + 1 = 0$ , what is the sum of all values of $x$ that satisfy the equation?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The sum of all values of

x

that satisfy the equation is 2.

By writing

16^{x-1}

as

\frac{(4^x)^2}{16}

and

10 \cdot 4^{x-2}

as

\frac{10 \cdot 4^x}{16}

, multiplying the entire equation by 16 yields

(4^x)^2 - 10(4^x) + 16 = 0

. Factoring this quadratic equation gives

(4^x - 2)(4^x - 8) = 0

, which yields

4^x = 2

and

4^x = 8

. Solving these exponential equations results in

x = 0.5

and

x = 1.5

. Adding these values together gives

0.5 + 1.5 = 2

.

Adım Adım Çözüm

1

Express each exponential term in the equation using a base of 4.

\frac{4^{2x}}{16} - \frac{10 \cdot 4^x}{16} + 1 = 0

Since

16 = 4^2

, we can write

16^{x-1} = (4^2)^{x-1} = 4^{2x-2} = \frac{4^{2x}}{16}

. Similarly,

4^{x-2} = \frac{4^x}{16}

.

2

Multiply every term in the equation by 16 to eliminate the fractions.

4^{2x} - 10 \cdot 4^x + 16 = 0

This simplifies the expression into a standard polynomial form that is quadratic in terms of

4^x

.

3

Use the substitution

u = 4^x

(where

u > 0

) to transform the equation into a quadratic equation in terms of

u

, and solve it.

u^2 - 10u + 16 = 0 \implies (u - 2)(u - 8) = 0 \implies u = 2

or

u = 8

Substituting

u

allows us to factor the quadratic expression using standard factoring techniques.

4

Substitute

4^x

back for

u

and solve for

x

using base 2 conversion.

For

u = 2

:

4^x = 2 \implies 2^{2x} = 2^1 \implies x = 0.5

. For

u = 8

:

4^x = 8 \implies 2^{2x} = 2^3 \implies x = 1.5

.

We must convert the solutions for the intermediate variable

u

back to the original variable

x

.

5

Calculate the sum of all found values of

x

.

0.5 + 1.5 = 2

The question asks for the sum of all real values of

x

that satisfy the equation.

Anahtar Kavram

Solving exponential equations of quadratic form by using variable substitution and exponent laws.

Advanced Math

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Alternatif Yöntem

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Cevap

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