Question

Let$f$be the linear function$f\left(x\right)=-5x+7$.

1. The rate of change of$f$is ______.
2. The graph of$f$is a _____ with slope _____ and$y$-intercept _____.

Short answer

1. The rate of change of $f$is $\underset{¯}{-5}$.
2. The graph of$f$ is a straight line with slope$\underset{¯}{-5}$ and $y$-intercept $\underset{¯}{7}$.

Step-by-step solution

Part a. Step 1. Definition of linear function.

In a linear function, each input value has a unique output value. In other words, for each point in domain, there is a unique value in the range of function. It can be written as:

$f\left(x\right)=ax+b$

Where$a$ is the rate of change (slope) and$b$ is the initial value ($y$-intercept), and both are real numbers.

Part a. Step 2. Given information.

It is given that$f\left(x\right)=-5x+7$ is a linear function.

The given statement is “The rate of change of$f$ is _______.”

Part a. Step 3. Find the required value for the blank.

In a linear function $f\left(x\right)=ax+b$, the coefficient of variable$x$ is the rate of change, i.e., $a$.

The given function is $f\left(x\right)=-5x+7$. Here the coefficient of variable$x$ is $-5$.

Thus, the rate of change of$f$ is $\underset{¯}{-5}$.

Part b. Step 1. Definition and graph of a linear function.

The rate of change of a linear function is constant and the graph of a linear function is always a straight line.

A linear function can be defined as:

$f\left(x\right)=ax+b$ … (i)

Where$a$ is the rate of change (slope) and$b$ is the initial value ($y$-intercept), and both are real numbers.

Part b. Step 2. Given information.

It is given that$f\left(x\right)=-5x+7$ is a linear function.

The given statement is “The graph of$f$ is a ____ with slope ____ and $y$-intercept ____.”

Part b. Step 3. Find the required values for the blanks.

Since the function$f$ is a linear function, therefore the graph of$f$ is a straight line.

Compare$f\left(x\right)=-5x+7$ with$f\left(x\right)=ax+b$.

$$\begin{gathered} a=-5 \\ b=7 \end{gathered}$$

Here, $a$is the slope and $b$is the $y$-intercept.

Thus, the graph of $f$ is a straight line with slope $\underset{¯}{-5}$and $y$-intercept $\underset{¯}{7}$.