# Chapter 4 - Section 4.1 - Properties of Linear Functions and Linear Models - 4.1 Assess Your Understanding: 13

(a) Slope $= 2$ and y-intercept $=3$ (b) See the image. (c) Average rate of change $= 2.0$ (d) The linear function $f(x)=2x+3$ is increasing.

#### Work Step by Step

Step-1: Compare the given equation with the point-slope form of the linear equation, that is, $f(x) = mx+b$, where $m$ is the slope of the linear function and $b$ is its y-intercept. By comparing $$f(x)=2x+3$$ to $$f(x) = mx+b$$ we understand that the slope of the given function is $2$ and its y-intercept is 3. Step-2: Let us put values of $x$ from $-2$ to $2$ into the function to obtain corresponding $y$ values. Using this we can plot a graph. Thus, For $x=-2$, $f(x)=-1$ For $x=-1$, $f(x)=1$ For $x=0$, $f(x)=3$ For $x=1$, $f(x)=5$ For $x=2$, $f(x) = 7$ This data yields the graph shown. Step-3: The average rate of change is defined as follows: $$\frac{\Delta y}{\Delta x}=\frac{f(x_2)-f(x_1)}{x_2-x_1}$$ Let us calculate average rate of change between $x_2=2$ and $x_1=-2$, $$\frac{\Delta y}{\Delta x}=\frac{7-(-1)}{2-(-2)}=2$$ Step-4: Since slope, $m=2>0$, this linear function is increasing.

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