Question

svlc algebra 1a - standard (15260)
linear functions
graph of two linear functions g(x) (blue) and f(x) (green) on a coordinate plane
how does the slope of g(x) compare to the slope of f(x)?

option 1: the slope of g(x) is greater than the slope of f(x).
option 2: the slope of g(x) is the opposite of the slope of f(x).
option 3: the slope of g(x) is less than the slope of f(x).
option 4: the slope of g(x) is equal to the slope of f(x).

Explanation:

Step1: Recall slope formula

The slope \( m \) of a line passing through two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \).

Step2: Find slope of \( g(x) \)

For line \( g(x) \), let's take two points. From the graph, when \( x = 0 \), \( y = 2 \) (point \((0,2)\)) and when \( x = 4 \), \( y = 4 \) (point \((4,4)\)). Using the slope formula:
\( m_{g}=\frac{4 - 2}{4 - 0}=\frac{2}{4}=\frac{1}{2} \)

Step3: Find slope of \( f(x) \)

For line \( f(x) \), take two points. When \( x = 0 \), \( y=- 2 \) (point \((0, - 2)\)) and when \( x = 2 \), \( y = 3 \) (point \((2,3)\)). Using the slope formula:
\( m_{f}=\frac{3-(-2)}{2 - 0}=\frac{5}{2} \)

Step4: Compare the slopes

We have \( m_{g}=\frac{1}{2} \) and \( m_{f}=\frac{5}{2} \). Since \( \frac{1}{2}<\frac{5}{2} \), the slope of \( g(x) \) is less than the slope of \( f(x) \).

Answer:

The slope of \( g(x) \) is less than the slope of \( f(x) \) (the option: "The slope of \( g(x) \) is less than the slope of \( f(x) \)").