Question

for a function g, suppose we know the following.
g(2) = -4, \quad g(2) = 1.5
write an equation of the tangent line to the graph of y = g(x) at x = 2.

Explanation:

Step1: Recall point-slope formula

The point-slope form of a line is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line, and $m$ is the slope.

Step2: Identify known values

We know $x_1=2$, $y_1=g(2)=-4$, and slope $m=g'(2)=1.5$. Substitute these into the formula:
$y - (-4) = 1.5(x - 2)$

Step3: Simplify the equation

Simplify the left side and rearrange to slope-intercept form (or keep in point-slope):
$y + 4 = 1.5x - 3$
$y = 1.5x - 7$

Answer:

$y = 1.5x - 7$ (or equivalent point-slope form $y + 4 = 1.5(x - 2)$)