The graph of linear function `g` passes through the points `\left(2,\ -1\right)`and `\left(-2,\ 7\right),`as shown. What are the slope and y-intercept of the graph `g`?

Answer 1

Given that the linear function \( g \) passes through the points \((2, -1)\) and \((-2, 7)\), we can find the slope \( m \) and the y-intercept \( b \) of the equation \( g(x) = mx + b \).

### Step 1: Calculate the Slope
The slope \( m \) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Given points:
\((x_1, y_1) = (2, -1)\)
\((x_2, y_2) = (-2, 7)\)

Substitute the values:
\[
m = \frac{7 - (-1)}{-2 - 2} = \frac{7 + 1}{-2 - 2} = \frac{8}{-4} = -2
\]

The slope \( m \) is \(-2\).

### Step 2: Find the Y-Intercept
The equation of the line in slope-intercept form is:
\[
g(x) = mx + b
\]

We substitute one of the given points, \((x_1, y_1) = (2, -1)\), and the slope \( m = -2 \) to solve for \( b \):
\[
-1 = -2(2) + b
\]
\[
-1 = -4 + b
\]
\[
b = -1 + 4
\]
\[
b = 3
\]

The y-intercept \( b \) is \( 3 \).

### Summary
The slope of the graph is \(-2\) and the y-intercept is \( 3 \).

Thus, the equation of the linear function \( g \) is:
\[
g(x) = -2x + 3
\]