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:
$$g(x)=-2x+3$$