Quadratic Functions and Graphs Alıştırma Soruları — SAT — Sayfa 4

Soru 61Soru →

In the $xy$ -plane, the graph of the quadratic function $f$ is a parabola with vertex $(4, -3)$ . If the graph passes through the point $(1, 15)$ , what is the value of $f(2)$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The value of

f(2)

is

5

.

By using the vertex form of a quadratic function,

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

with vertex

(4, -3)

, the function can be written as

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

. Substituting the point

(1, 15)

gives

15 = a(1 - 4)^2 - 3

, which simplifies to

18 = 9a

, leading to

a = 2

. Substituting

a = 2

back into the function gives

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

. Finally, evaluating at

x = 2

gives

f(2) = 2(2 - 4)^2 - 3 = 5

.

Adım Adım Çözüm

1

Write the quadratic function in vertex form and substitute the vertex

(4, -3)

.

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

The vertex form of a quadratic function is

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

, where

(h, k)

is the vertex.

2

Substitute the point

(1, 15)

into the equation and solve for the constant

a

.

a = 2

Since the graph passes through

(1, 15)

, substituting

x = 1

and

f(x) = 15

allows us to solve for

a

.

3

Substitute

x = 2

into the completed function

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

to find

f(2)

.

f(2) = 5

Evaluating the function at

x = 2

yields the required value.

Anahtar Kavram

Determining a quadratic function's equation from its vertex and a point, then evaluating it.

Soru 62Soru →

The trajectory of a basketball thrown toward a hoop can be modeled by a quadratic function. In the $xy$ -plane, $x$ represents the horizontal distance in feet from the shooter and $y$ represents the height of the basketball in feet. The basketball reaches its maximum height of $14$ feet at a horizontal distance of $8$ feet from the shooter. If the basketball is released at a height of $6$ feet, which of the following equations models the trajectory of the basketball?

y = -\frac{1}{8}(x + 8)^2 + 14

y = -\frac{1}{8}(x - 14)^2 + 8

y = -\frac{1}{8}(x - 8)^2 + 14

y = -8(x - 8)^2 + 14

Cevabı ve açıklamayı göster

Cevap: $y = -\frac{1}{8}(x - 8)^2 + 14$

Cevap

The equation

y = -\frac{1}{8}(x - 8)^2 + 14

The vertex form of a quadratic function is

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

, where

(h, k)

represents the coordinates of the vertex. Since the maximum height of the basketball is

14

feet at a horizontal distance of

8

feet, the vertex is

(8, 14)

. Substituting these values gives the equation

y = a(x - 8)^2 + 14

. The release point represents the y-intercept where

x = 0

and

y = 6

. Substituting these coordinates into the equation gives

6 = a(0 - 8)^2 + 14

, which simplifies to

6 = 64a + 14

. Subtracting

14

from both sides results in

-8 = 64a

, and dividing both sides by

64

yields

a = -\frac{1}{8}

. Therefore, the correct trajectory model is

y = -\frac{1}{8}(x - 8)^2 + 14

.

Adım Adım Çözüm

1

Identify the vertex coordinates

(h, k)

of the parabolic path from the given maximum height context.

The vertex

(h, k)

is

(8, 14)

.

The basketball reaches its maximum height of

14

feet at a horizontal distance of

8

feet, and the maximum of a downward-opening parabola is its vertex.

2

Write the general vertex form of a quadratic equation and substitute the coordinates of the vertex.

y = a(x - 8)^2 + 14

The vertex form of a quadratic equation is

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

, where

(h, k)

is the vertex.

3

Substitute the initial release point

(0, 6)

into the equation to solve for the coefficient

a

.

6 = a(0 - 8)^2 + 14 \implies 6 = 64a + 14 \implies -8 = 64a \implies a = -\frac{8}{64} = -\frac{1}{8}

The basketball is released at a height of

6

feet when the horizontal distance

x = 0

, representing the y-intercept of the trajectory.

4

Substitute the solved value of

a

back into the vertex form equation.

y = -\frac{1}{8}(x - 8)^2 + 14

This yields the complete quadratic model representing the path of the basketball.

Anahtar Kavram

Writing quadratic equations in vertex form from context

Soru 63Soru →

In the $xy$ -plane, the graph of the quadratic function $f$ is a parabola with vertex $(3, -8)$ and passes through the point $(1, 0)$ . If the graph intersects the $y$ -axis at $(0, c)$ , what is the value of $c$ ?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

The correct answer is 10.

By writing the function in vertex form

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

and substituting the vertex

(3, -8)

, we obtain

f(x) = a(x - 3)^2 - 8

. Substituting the point

(1, 0)

into the equation gives

0 = a(1 - 3)^2 - 8

, which simplifies to

4a = 8

and yields

a = 2

. With the function fully defined as

f(x) = 2(x - 3)^2 - 8

, the

y

-intercept is found by calculating

f(0) = 2(0 - 3)^2 - 8 = 2(9) - 8 = 10

. Thus,

c = 10

.

Adım Adım Çözüm

1

Express the quadratic function in vertex form using the given vertex

(3, -8)

.

f(x) = a(x - 3)^2 - 8

The vertex form of a quadratic function is

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

, where

(h, k)

is the vertex.

2

Substitute the coordinates of the point

(1, 0)

into the function to solve for the leading coefficient

a

.

a = 2

Since the graph passes through

(1, 0)

, substituting

x = 1

and

f(x) = 0

must satisfy the equation.

3

Substitute

x = 0

into the completed function

f(x) = 2(x - 3)^2 - 8

to find the value of

c

.

c = 10

The intersection with the

y

-axis occurs where the input

x

is 0.

Anahtar Kavram

Writing and evaluating quadratic functions using the vertex form

Soru 64Soru →

In the $xy$ -plane, the graph of the quadratic function $f$ is defined by $f(x) = -x^2 + 6x - 5$ . If the graph of $f$ is translated $2$ units to the right and $4$ units up to form the graph of a new function $g$ , what is the maximum value of $g$ ?

0

5

8

10

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

8

The maximum value of the quadratic function is the y-coordinate of its vertex. For

f(x) = -x^2 + 6x - 5

, the vertex occurs at

x = -\frac{6}{2(-1)} = 3

. Evaluating

f(3)

yields

-9 + 18 - 5 = 4

, which is the maximum value of

f

. Translating the graph

2

units to the right shifts the vertex horizontally to

x = 5

but does not change the maximum height. Translating the graph

4

units up increases the y-coordinate of the vertex by

4

, resulting in a new maximum value of

4 + 4 = 8

.

Adım Adım Çözüm

1

Find the x-coordinate of the vertex of the function

f(x) = -x^2 + 6x - 5

.

x = 3

The x-coordinate of the vertex of a quadratic function in the standard form

y = ax^2 + bx + c

is given by

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

. For

f(x)

, we have

a = -1

and

b = 6

, so

x = -\frac{6}{2(-1)} = 3

.

2

Calculate the maximum value of

f(x)

by finding the y-coordinate of its vertex.

f(3) = 4

Substitute

x = 3

into the function:

f(3) = -(3)^2 + 6(3) - 5 = -9 + 18 - 5 = 4

. Since the coefficient of

x^2

is negative, the parabola opens downward, meaning the vertex represents the maximum value.

3

Apply the vertical translation to find the maximum value of the new function

g(x)

.

Maximum value of

g

is

8

Translating the graph

2

units to the right shifts the vertex horizontally but does not affect the vertical maximum value. Translating the graph

4

units up increases the maximum value by

4

. Thus, the new maximum value is

4 + 4 = 8

.

Anahtar Kavram

Determining the maximum value of a quadratic function and applying translations to its graph.

Soru 65Soru →

The graph of the quadratic equation $y = 2x^2 - 12x + c$ , where $c$ is a constant, is a parabola in the $xy$ -plane. If the $y$ -coordinate of the vertex of this parabola is $-5$ , what is the value of $c$ ?

Cevabı ve açıklamayı göster

Cevap: 13

Cevap

13

To find the value of the constant

c

, we calculate the coordinates of the vertex of the parabola. The x-coordinate of the vertex for a quadratic function in standard form

y = ax^2 + bx + c

is given by

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

. Substituting

a = 2

and

b = -12

gives

x = -\frac{-12}{2(2)} = 3

. Evaluating the quadratic equation at

x = 3

gives the y-coordinate of the vertex:

y = 2(3)^2 - 12(3) + c = 18 - 36 + c = c - 18

. Since we are given that the y-coordinate of the vertex is

-5

, we set

c - 18 = -5

and solve to find

c = 13

.

Adım Adım Çözüm

1

Calculate the x-coordinate of the vertex using the vertex formula.

x = 3

The axis of symmetry and the x-coordinate of the vertex are located at

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

.

2

Substitute the x-coordinate of the vertex into the equation to express the y-coordinate in terms of

c

.

y = c - 18

Evaluating the quadratic function at the vertex's x-coordinate gives the minimum or maximum value of the function.

3

Equate the expression for the y-coordinate to the given vertex y-coordinate of

-5

and solve for

c

.

c = 13

Setting the calculated y-coordinate expression equal to the given value allows us to isolate and solve for the unknown constant.

Anahtar Kavram

Determining the vertex of a quadratic function from its standard form and solving for a constant coefficient.

Alternatif Yöntem

Alternatively, we can complete the square to write the quadratic equation in vertex form,

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

. Factoring the leading coefficient from the variable terms gives

y = 2(x^2 - 6x) + c

. To complete the square inside the parentheses, add and subtract

9

:

y = 2(x^2 - 6x + 9 - 9) + c = 2((x-3)^2 - 9) + c = 2(x-3)^2 - 18 + c

. In this vertex form, the y-coordinate of the vertex is

k = c - 18

. Since the vertex y-coordinate is

-5

, we set

c - 18 = -5

to get

c = 13

.

Tahmini Süre:1m 30s

Soru 66Soru →

The graph of the quadratic function $h$ has a vertex at $(2, -5)$ in the $xy$ -plane. If the function $g$ is defined by $g(x) = h(x + 3) + 4$ , what is the vertex of the graph of $g$ ?

(-1, -1)

(5, -1)

(-5, -1)

(1, -1)

Cevabı ve açıklamayı göster

Cevap: $(-1, -1)$

Cevap

The vertex of the graph of

g

is

(-1, -1)

.

The vertex of the graph of

h

is given as

(2, -5)

. The function

g(x) = h(x + 3) + 4

represents a transformation of the graph of

h

. Specifically,

h(x + 3)

shifts the graph horizontally to the left by

3

units, and adding

4

shifts the graph vertically up by

4

units. Applying these shifts to the original vertex coordinates: the new

x

-coordinate is

2 - 3 = -1

, and the new

y

-coordinate is

-5 + 4 = -1

. Therefore, the vertex of the graph of

g

is

(-1, -1)

.

Adım Adım Çözüm

1

Identify the vertex of the original function

h

.

The vertex of

h

is

(2, -5)

.

The problem states that the graph of the quadratic function

h

has a vertex at

(2, -5)

.

2

Determine the horizontal translation from the expression

h(x + 3)

.

A horizontal shift of

3

units to the left.

Replacing

x

with

x + c

in a function

f(x)

shifts the graph left by

c

units. Here,

c = 3

, so we shift the vertex

x

-coordinate left by

3

:

2 - 3 = -1

.

3

Determine the vertical translation from the addition of

4

.

A vertical shift of

4

units up.

Adding a constant

d

to a function shifts its graph vertically by

d

units. Here,

d = 4

, so we shift the vertex

y

-coordinate up by

4

:

-5 + 4 = -1

.

4

Combine the shifted coordinates to find the new vertex.

The new vertex is

(-1, -1)

.

Combining the translated

x

-coordinate of

-1

and

y

-coordinate of

-1

gives the vertex of the graph of

g

.

Anahtar Kavram

Quadratic Function Transformations

Soru 67Soru →

A quadratic function $f$ is defined by $f(x) = a(x - 4)(x - k)$ , where $a$ and $k$ are constants. In the $xy$ -plane, the graph of $y = f(x)$ has its vertex at $(6, 12)$ . What is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The value of

k

is

8

.

The given quadratic function is in the factored form

f(x) = a(x - 4)(x - k)

, meaning the x-intercepts of its graph are

x = 4

and

x = k

. Due to the symmetry of a parabola, the x-coordinate of the vertex is the midpoint of the x-intercepts. Since the vertex is at

(6, 12)

, its x-coordinate is

6

. Setting the midpoint of the intercepts equal to 6 gives the equation

\frac{4 + k}{2} = 6

. Multiplying both sides by 2 gives

4 + k = 12

, and subtracting 4 gives the correct answer

k = 8

.

Adım Adım Çözüm

1

Identify the x-intercepts of the function

f(x) = a(x - 4)(x - k)

.

The x-intercepts are at

x = 4

and

x = k

.

For any quadratic function written in the factored form

f(x) = a(x - r_1)(x - r_2)

, the values

r_1

and

r_2

correspond to the x-intercepts of the graph.

2

Relate the x-coordinate of the vertex to the x-intercepts using symmetry.

The axis of symmetry equation is

\frac{4 + k}{2} = 6

.

Because a parabola is symmetric, the x-coordinate of the vertex always lies exactly halfway between the two x-intercepts.

3

Solve the equation for

k

.

k = 8

.

Multiplying both sides of the equation

\frac{4 + k}{2} = 6

by 2 yields

4 + k = 12

. Subtracting 4 from both sides gives

k = 8

.

Anahtar Kavram

Symmetry of quadratic graphs and their vertices relative to their x-intercepts.

Alternatif Yöntem

Alternatively, substitute the vertex coordinates

(6, 12)

into the function:

12 = a(6 - 4)(6 - k) = 2a(6 - k)

. Since the vertex is the maximum point, the derivative

f'(x) = a(x - k) + a(x - 4)

must be equal to 0 at

x = 6

. This yields

a(6 - k) + a(6 - 4) = 0

. Since

a \neq 0

, dividing by

a

gives

(6 - k) + 2 = 0

, which simplifies to

8 - k = 0

, or

k = 8

.

Tahmini Süre:1m 30s

Soru 68Soru →

The graph of the quadratic function $f$ in the $xy$ -plane is a parabola. If $f(-3) = -2$ and $f(7) = -2$ , what is the $x$ -coordinate of the vertex of the parabola?

4

2

-2

5

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The correct answer is 2, since the vertex lies on the axis of symmetry, which is the midpoint of the symmetric x-coordinates.

The vertex of a parabola in the

xy

-plane lies on its axis of symmetry. For any quadratic function, points with the same

y

-coordinate are symmetric with respect to the axis of symmetry. Since

f(-3) = -2

and

f(7) = -2

, the

y

-coordinates are equal, which means the axis of symmetry is the vertical line midway between

x = -3

and

x = 7

. The

x

-coordinate of the vertex is the midpoint of these two

x

-values, calculated as

\frac{-3 + 7}{2} = 2

.

Adım Adım Çözüm

1

Identify the two points on the parabola with the same y-value.

The points are

(-3, -2)

and

(7, -2)

.

Since both points share the same y-value of

-2

, they are symmetric with respect to the axis of symmetry of the parabola.

2

Find the x-coordinate of the axis of symmetry by calculating the midpoint of the x-coordinates of the two symmetric points.

\text{Midpoint} = \frac{-3 + 7}{2} = \frac{4}{2} = 2

.

The axis of symmetry is always located exactly halfway between any two symmetric points on the parabola.

3

Determine the x-coordinate of the vertex of the parabola.

The x-coordinate of the vertex is 2.

The vertex of a parabola lies on its axis of symmetry, so its x-coordinate is equal to the x-coordinate of the axis of symmetry.

Anahtar Kavram

Symmetry of quadratic graphs and the axis of symmetry

Alternatif Yöntem

Alternatively, any quadratic function that takes the value

-2

at

x = -3

and

x = 7

can be written in the form

f(x) = a(x + 3)(x - 7) - 2

for some constant

a

. Expanding this expression yields

f(x) = a(x^2 - 4x - 21) - 2

. Since the

x

-coordinate of the vertex of a quadratic function in standard form

y = ax^2 + bx + c

is given by

-\frac{b}{2a}

, we can find the vertex

x

-coordinate as

-\frac{-4a}{2a} = 2

.

Tahmini Süre:1m 30s

Soru 69Soru →

In the $xy$ -plane, the graph of the quadratic function $f$ , defined by $f(x) = -(x - 4)^2 + c$ where $c$ is a constant, intersects the $x$ -axis at two points. If the distance between these two points is $10$ , what is the value of $c$ ?

Cevabı ve açıklamayı göster

Cevap: 25

Cevap

25

The quadratic function is defined by

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

, which is in the vertex form

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

. The vertex of this parabola is at

(4, c)

, and the vertical line

x = 4

is its axis of symmetry. The distance between the two

x

-intercepts is

10

. By symmetry, the intercepts must be located

5

units to the left and right of the axis of symmetry, placing them at

x = 4 - 5 = -1

and

x = 4 + 5 = 9

. Since the graph intersects the

x

-axis at these points, substituting either

x

-coordinate into the function gives a

y

-value of

0

. Substituting

x = 9

yields

0 = -(9 - 4)^2 + c

, which simplifies to

0 = -25 + c

, so

c = 25

.

Adım Adım Çözüm

1

Identify the axis of symmetry of the quadratic function.

The axis of symmetry is

x = 4

.

The function

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

is written in vertex form,

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

, where the vertex is

(h, k)

and the axis of symmetry is

x = h

.

2

Find the

x

-coordinates of the

x

-intercepts using the distance between them.

The

x

-intercepts are at

x = -1

and

x = 9

.

Since the parabola is symmetric about the line

x = 4

and the distance between the two intercepts is

10

, each intercept is

\frac{10}{2} = 5

units away from the axis of symmetry. Thus, the intercepts are at

4 - 5 = -1

and

4 + 5 = 9

.

3

Substitute one of the

x

-intercepts into the function to solve for the constant

c

.

c = 25

Since

(9, 0)

is on the graph,

f(9) = 0

. Substituting this gives

0 = -(9 - 4)^2 + c \implies 0 = -25 + c \implies c = 25

.

Anahtar Kavram

Using symmetry and the vertex form of a quadratic function to determine constants from key graphical features such as

x

-intercepts.

Soru 70Soru →

The graph of the quadratic function $f$ in the $xy$ -plane is defined by $f(x) = x^2 - 6x + 5$ . If the graph of $f$ is translated $3$ units to the left and $2$ units down, which of the following equations defines the resulting function $g$ ?

g(x) = x^2 - 12x + 30

g(x) = x^2 - 6

g(x) = x^2 - 2

g(x) = x^2 - 12x + 34

Cevabı ve açıklamayı göster

Cevap: $g(x) = x^2 - 6$

Cevap

The equation that defines the function

g

is

g(x) = x^2 - 6

.

To find the function

g(x)

, we apply the transformations to

f(x) = x^2 - 6x + 5

. A translation of

3

units to the left replaces

x

with

x + 3

, which gives

f(x + 3) = (x + 3)^2 - 6(x + 3) + 5 = x^2 + 6x + 9 - 6x - 18 + 5 = x^2 - 4

. Next, translating the graph

2

units down corresponds to subtracting

2

from the function, resulting in

g(x) = (x^2 - 4) - 2 = x^2 - 6

.

Adım Adım Çözüm

1

Rewrite the function

f(x)

in vertex form by completing the square.

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

Converting to vertex form

a(x - h)^2 + k

makes it straightforward to apply horizontal and vertical shifts based on the vertex

(h, k)

.

2

Apply the horizontal translation of

3

units to the left by replacing

x

with

x + 3

in the function.

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

A horizontal shift of

c

units to the left is represented by replacing

x

with

x + c

.

3

Apply the vertical translation of

2

units down by subtracting

2

from the expression obtained in the previous step.

g(x) = (x^2 - 4) - 2 = x^2 - 6

A vertical shift of

d

units downward is represented by subtracting

d

from the function's output.

Anahtar Kavram

Translations of quadratic functions in the coordinate plane.

Tahmini Süre:1m 30s

Soru 71Soru →

In the $xy$ -plane, the vertex of the parabola defined by $y = a(x - 2)(x - 8)$ has a $y$ -coordinate of $-18$ , where $a$ is a positive constant. What is the value of $a$ ?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

2

The correct answer is 2. The quadratic function is given in factored form as

y = a(x - 2)(x - 8)

. The x-intercepts of this parabola are at

x = 2

and

x = 8

. Because of the symmetry of a parabola, the x-coordinate of the vertex is the midpoint of the x-intercepts:

x = \frac{2 + 8}{2} = 5

. The y-coordinate of the vertex is given as

-18

, meaning the vertex is at the point

(5, -18)

. Substituting these coordinates into the equation gives

-18 = a(5 - 2)(5 - 8)

, which simplifies to

-18 = a(3)(-3) = -9a

. Solving for

a

yields

a = 2

.

Adım Adım Çözüm

1

Identify the x-intercepts from the factored form equation

y = a(x - 2)(x - 8)

and find the x-coordinate of the vertex.

The x-intercepts are

x = 2

and

x = 8

. The x-coordinate of the vertex is the midpoint of the intercepts:

x = \frac{2 + 8}{2} = 5

.

The axis of symmetry of a parabola passes through its vertex and lies midway between its x-intercepts.

2

Substitute the coordinates of the vertex

(5, -18)

into the quadratic equation to solve for the constant

a

.

Substituting

x = 5

and

y = -18

yields

-18 = a(5 - 2)(5 - 8)

, which simplifies to

-18 = a(3)(-3)

, so

-18 = -9a

, giving

a = 2

.

Since the vertex is a point on the parabola, its coordinates must satisfy the equation of the parabola.

Anahtar Kavram

Using the symmetry of quadratic functions in factored form to find the vertex coordinates.

Soru 72Soru →

The graph of the quadratic function $f$ in the $xy$ -plane has its vertex at $(3, 2)$ . If the graph passes through the point $(5, -6)$ , what is the $y$ -intercept of the graph of $f$ ?

-16

-11

-1

2

Cevabı ve açıklamayı göster

Cevap: $-16$

Cevap

The y-intercept of the graph of

f

is

-16

.

The correct answer is found by setting up the vertex form of the quadratic function,

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

, using the vertex

(3, 2)

. Substituting the point

(5, -6)

yields

-6 = a(2)^2 + 2

, which simplifies to

-8 = 4a

, giving

a = -2

. The complete equation is

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

. Evaluating this function at

x = 0

yields

-2(0-3)^2 + 2 = -18 + 2 = -16

.

Adım Adım Çözüm

1

Write the quadratic function in vertex form.

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

The vertex form of a quadratic function is

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

, where

(h, k)

is the vertex. Here,

(h, k) = (3, 2)

.

2

Substitute the coordinates of the given point

(5, -6)

to solve for the coefficient

a

.

a = -2

Plugging

x=5

and

f(x)=-6

into the equation gives

-6 = a(5-3)^2 + 2

, which simplifies to

-6 = 4a + 2

. Subtracting

2

from both sides gives

-8 = 4a

, so

a = -2

.

3

Substitute

a = -2

back into the vertex form to write the full equation of the function.

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

This represents the specific quadratic function described.

4

Evaluate the function at

x = 0

to find the y-intercept.

f(0) = -16

The y-intercept of a graph is the point where

x=0

. Evaluating

f(0)

gives

-2(0-3)^2 + 2 = -2(9) + 2 = -16

.

Anahtar Kavram

Writing a quadratic function in vertex form

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

and finding the y-intercept by evaluating the function at

x=0

.

Tahmini Süre:1m 30s

Soru 73Soru →

The function $h(t) = -16t^2 + v_0 t + h_0$ models the height $h(t)$ , in feet, of a model rocket $t$ seconds after launch, where $v_0$ and $h_0$ are constants. The rocket reaches its maximum height of $100$ feet above the ground $2$ seconds after it is launched. Which of the following equations defines $h(t)$ ?

h(t) = -16t^2 + 64t + 36

h(t) = -16t^2 - 64t + 36

h(t) = -16t^2 + 64t + 100

h(t) = -16t^2 + 64t - 164

Cevabı ve açıklamayı göster

Cevap: $h(t) = -16t^2 + 64t + 36$

Cevap

h(t) = -16t^2 + 64t + 36

The correct equation is

h(t) = -16t^2 + 64t + 36

. Since the maximum height of

100

feet is reached at

t = 2

seconds, the vertex of the parabola is

(2, 100)

. In vertex form, a quadratic function is written as

h(t) = a(t - d)^2 + c

, where

(d, c)

is the vertex. Given that the leading coefficient

a

is

-16

, substituting the vertex yields

h(t) = -16(t - 2)^2 + 100

. Expanding this expression gives

h(t) = -16(t^2 - 4t + 4) + 100 = -16t^2 + 64t - 64 + 100 = -16t^2 + 64t + 36

.

Adım Adım Çözüm

1

Identify the vertex from the problem description.

The vertex of the parabola is

(2, 100)

, representing the time

t = 2

seconds when the maximum height of

100

feet is reached.

The vertex

(h, k)

of a quadratic function represents the maximum or minimum point of its graph.

2

Write the quadratic equation in vertex form.

h(t) = a(t - 2)^2 + 100

. Since the coefficient of

t^2

in the standard form is

-16

, we set

a = -16

, giving

h(t) = -16(t - 2)^2 + 100

.

The vertex form of a quadratic function is

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

, and the leading coefficient

a

is the same as in standard form.

3

Expand the vertex form equation into standard form.

h(t) = -16(t^2 - 4t + 4) + 100 = -16t^2 + 64t - 64 + 100 = -16t^2 + 64t + 36

.

Expanding the squared term and distributing the leading coefficient converts the vertex form to standard form

y = ax^2 + bx + c

.

Anahtar Kavram

Quadratic Functions and Graphs

Alternatif Yöntem

The axis of symmetry for a quadratic function in standard form

y = ax^2 + bx + c

is given by

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

. For this model, the maximum occurs at

t = 2

, meaning the axis of symmetry is

t = 2

. Since

a = -16

, we have

2 = -\frac{b}{2(-16)}

, which simplifies to

b = 64

. We can then test the remaining options where the linear coefficient is

64

. Substituting

t = 2

into the correct equation

h(t) = -16t^2 + 64t + 36

yields

h(2) = -16(4) + 64(2) + 36 = -64 + 128 + 36 = 100

, confirming it reaches the correct maximum height.

Tahmini Süre:1m 30s

Soru 74Soru →

In the $xy$ -plane, the graph of the quadratic function $f$ has a vertex at $(h, k)$ , where $h$ and $k$ are constants. The graph passes through the points $(1, 10)$ and $(7, 10)$ . If the minimum value of the function $f$ is $1$ , what is the value of $f(2)$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The value of

f(2)

is 5.

The correct answer is 5. Since the graph of the quadratic function passes through

(1, 10)

and

(7, 10)

, the axis of symmetry is the vertical line halfway between

x = 1

and

x = 7

, which is

x = \frac{1+7}{2} = 4

. The minimum value of the function is

1

, which occurs at the vertex, so the vertex is

(4, 1)

. In vertex form, the function is

f(x) = a(x-4)^2 + 1

. Substituting the point

(1, 10)

yields

10 = a(1-4)^2 + 1

, which simplifies to

9 = 9a

, so

a = 1

. The function is

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

. Evaluating this function at

x = 2

gives

f(2) = (2-4)^2 + 1 = 5

.

Adım Adım Çözüm

1

Find the

x

-coordinate of the vertex using the symmetry of the parabola.

h = 4

A parabola is symmetric about its vertical axis of symmetry. Since the points

(1, 10)

and

(7, 10)

have the same

y

-coordinate, the axis of symmetry is exactly halfway between their

x

-coordinates:

x = \frac{1 + 7}{2} = 4

.

2

Find the vertex coordinates using the minimum value of the function.

Vertex is

(4, 1)

The vertex of a quadratic function with a minimum value lies on the axis of symmetry, and its

y

-coordinate is the minimum value. Thus, the vertex

(h, k)

is

(4, 1)

.

3

Write the vertex form of the quadratic function and solve for the leading coefficient

a

.

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

Using the vertex form

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

, we substitute

h = 4

and

k = 1

to get

f(x) = a(x - 4)^2 + 1

. Substituting the point

(1, 10)

gives

10 = a(1 - 4)^2 + 1 \implies 9 = 9a \implies a = 1

.

4

Evaluate

f(2)

using the determined function.

f(2) = 5

Substitute

x = 2

into the equation

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

to obtain

f(2) = (2 - 4)^2 + 1 = 4 + 1 = 5

.

Anahtar Kavram

Using symmetry and the vertex form of a quadratic function to determine its equation and evaluate values.

Tahmini Süre:1m 30s

Soru 75Soru →

The quadratic function $f$ is defined by $f(x) = (x - 4)(x - 10)$ . In the $xy$ -plane, the graph of function $g$ is obtained by translating the graph of $f$ horizontally so that the vertex of the graph of $g$ lies on the $y$ -axis. Which of the following equations defines the function $g$ ?

g(x) = x^2 - 14x + 40

g(x) = x^2 - 9

g(x) = x^2 - 28x + 187

g(x) = x^2 + 14x + 40

Cevabı ve açıklamayı göster

Cevap: $g(x) = x^2 - 9$

Cevap

The correct equation is

g(x) = x^2 - 9

.

The x-coordinate of the vertex of the graph of

f(x) = (x - 4)(x - 10)

is the average of its x-intercepts:

\frac{4 + 10}{2} = 7

. For the vertex of the graph of

g

to lie on the y-axis, the x-coordinate of the new vertex must be 0. This requires translating the graph of

f

to the left by 7 units, which corresponds to the transformation

g(x) = f(x + 7)

. Substituting

x + 7

for

x

in

f(x)

gives

g(x) = ((x + 7) - 4)((x + 7) - 10) = (x + 3)(x - 3) = x^2 - 9

.

Adım Adım Çözüm

1

Identify the x-intercepts of the function

f(x) = (x - 4)(x - 10)

.

The x-intercepts are

x = 4

and

x = 10

.

The x-intercepts occur where the function equals zero, which are the roots of the factors.

2

Find the x-coordinate of the vertex of the graph of

f

.

The x-coordinate of the vertex is

x = 7

.

The axis of symmetry (and thus the x-coordinate of the vertex) is the average of the x-intercepts:

\frac{4 + 10}{2} = 7

.

3

Determine the translation needed to place the vertex on the y-axis.

Translate the graph to the left by 7 units.

The y-axis corresponds to

x = 0

. To move the vertex from

x = 7

to

x = 0

, the graph must be shifted 7 units to the left.

4

Apply the horizontal translation to find the equation for

g(x)

.

g(x) = (x + 3)(x - 3) = x^2 - 9

.

A horizontal translation of 7 units to the left is represented by

g(x) = f(x + 7)

. Substituting

x + 7

for

x

in the equation for

f(x)

gives

((x + 7) - 4)((x + 7) - 10) = (x + 3)(x - 3) = x^2 - 9

.

Anahtar Kavram

Identifying the vertex of a quadratic function from its factored form and applying horizontal translation rules.

Soru 76Soru →

In the $xy$ -plane, the graph of $y = x^2 - 8x + 12$ represents the quadratic function $f$ . If this graph is shifted $3$ units to the left and $5$ units up to create the graph of a new function $g$ , what is the $y$ -value of the vertex of the graph of $g$ ?

Cevabı ve açıklamayı göster

Cevap: 1

Cevap

The y-value of the vertex of the graph of g is 1.

The vertex of the original quadratic function f(x) = x^2 - 8x + 12 can be found by rewriting it in vertex form, which is f(x) = (x - 4)^2 - 4. This shows that the vertex of the original graph is (4, -4). A translation of 3 units to the left subtracts 3 from the x-coordinate of the vertex (4 - 3 = 1), and a translation of 5 units up adds 5 to the y-coordinate of the vertex (-4 + 5 = 1). Thus, the vertex of the graph of the new function g is (1, 1), making its y-value 1.

Adım Adım Çözüm

1

Find the vertex of the original quadratic function.

The vertex of the original graph is

(4, -4)

.

By completing the square on

y = x^2 - 8x + 12

, we get

y = (x - 4)^2 - 4

, which reveals the vertex is at

(4, -4)

.

2

Apply the translation to the vertex coordinates.

The translated vertex coordinates are

(1, 1)

.

Shifting a point

(x, y)

by

3

units left and

5

units up results in the point

(x - 3, y + 5)

. Applying this to the vertex

(4, -4)

yields

(4 - 3, -4 + 5) = (1, 1)

.

3

Identify the y-coordinate of the new vertex.

The y-value is

1

.

The vertex of the graph of

g

is

(1, 1)

, where the second coordinate represents the y-value.

Anahtar Kavram

Identifying the vertex of a quadratic function and applying horizontal and vertical translations in the coordinate plane.

Quadratic Functions and Graphs

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