Question 217·Medium·Algebra

Use the slope formula $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ with the two points $(2,-3)$ and $(8,15)$. Be careful with the negative sign in $-3$.

Once you know the slope, use point-slope form $y - y_1 = m(x - x_1)$ with one of the points to write $g(x)$ as an expression in $x$.

After you have an equation for $g(x)$, set it equal to 9 (because $g(x)=9$) and solve that linear equation for $x$.

In Desmos, enter the equation in point-slope form: y + 3 = ((15 - (-3))/(8 - 2))(x - 2). This graphs the unique line passing through $(2,-3)$ and $(8,15)$.

On a new line, type y = 9 to draw a horizontal line where the function value is 9.

Click on the intersection point of the two lines. The x-coordinate of this intersection is the value of $x$ that makes $g(x)=9$.

A linear function’s graph is a straight line. A line is completely determined by any two points on it, so the information $g(2)=-3$ and $g(8)=15$ lets us find the equation of $g$.

Treat the function values as points: $(2,-3)$ and $(8,15)$. The slope $m$ is

$m = \dfrac{15-(-3)}{8-2} = \dfrac{15+3}{6} = \dfrac{18}{6} = 3$.

So the slope of $g$ is 3.

Use point-slope form with point $(2,-3)$ and slope $3$:

which simplifies to

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

Subtract 3 from both sides or distribute to get slope-intercept form:

$g(x) = 3x - 9$.

We want $g(x) = 9$. Substitute into the equation:

Add 9 to both sides:

Divide both sides by 3:

$x = 6$.

So $g(x) = 9$ when $x=6$. This is the correct answer.