In the xy-plane, the graph of the linear function f passes through the point (2, 5) and ha… — SAT

In the $xy$ -plane, the graph of the linear function $f$ passes through the point $(2, 5)$ and has a $y$ -intercept of $(0, b)$ , where $b$ is a constant. The function $g$ is defined by $g(x) = f(x) - 4$ . If the $x$ -intercept of the graph of $g$ is $(6, 0)$ , what is the value of $b$ ?

A
10
B
$\frac{19}{2}$
$\frac{11}{2}$ Answer
D
$\frac{11}{4}$

Answer

The correct value of

b

\frac{11}{2}

The correct answer is the value

\frac{11}{2}

. By writing

f(x) = mx + b

and using the point

(2, 5)

, we get

m = \frac{5-b}{2}

. Applying the definition of

g(x) = f(x) - 4

gives

g(x) = mx + b - 4

. Since the graph of

g

has an

x

-intercept of

(6, 0)

, substituting

x = 6

and

g(6) = 0

gives

6m + b - 4 = 0

. Substituting the expression for

m

yields

3(5-b) + b - 4 = 0

, which simplifies to

11 - 2b = 0

, and thus

b = \frac{11}{2}

Step-by-Step Solution

Write the linear function

f(x)

in slope-intercept form using the given

y

-intercept

(0, b)

f(x) = mx + b

, where

m

is the slope of the line.

This sets up the general equation of the line with unknown parameters

m

and

b

Substitute the coordinates of the point

(2, 5)

into the equation for

f(x)

to express

m

in terms of

b

5 = 2m + b

, which simplifies to

m = \frac{5 - b}{2}

Since the point lies on the graph of

f

, its coordinates must satisfy the equation.

Define

g(x)

using the relationship

g(x) = f(x) - 4

and apply the

x

-intercept

(6, 0)

g(x) = mx + b - 4

. Since the

x

-intercept is

(6, 0)

, we have

g(6) = 0

, which gives

6m + b - 4 = 0

The

x

-intercept is the point where the output of the function is zero.

Substitute the expression for

m

from Step 2 into the equation from Step 3 and solve for

b

6\left(\frac{5-b}{2}\right) + b - 4 = 0 \Rightarrow 3(5-b) + b - 4 = 0 \Rightarrow 15 - 3b + b - 4 = 0 \Rightarrow 11 - 2b = 0 \Rightarrow b = \frac{11}{2}

This solves the system of equations to determine the value of the constant

b

Key Concept

Linear Functions and Graphs

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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