Question

Find an equation of the tangent line to the graph of

y = g(x) at x = 5 if g(5) = −4 and g'(5) = 3.
(Enter your answer as an equation in terms of y and x.)

Answer 1

y + 4 = -3 (x - 5)

In other words,

y = -3 x + 11

Explanation:

The slope of the tangent line to y = g(x) at x = 5 is the same as the value of g'(x). g'(5) = 3. Therefore, 3 will be the slope of the tangent line.

The tangent line goes through the point of tangency (5, g(5)). g(5) = -4. Therefore, the tangent line passes through the point (5, -4).

Apply the slope-point form of the line. The equation for a line with slope m that goes through point (a, b) will be y - b = m(x - a). For the tangent line in this question,

• m = 3,
• a = 5, and
• b = -4.

What will be the equation of this line?

Answer 2

Answer: y = 3x - 19

Explanation:

y = g(x)

g(5) = -4

⇒ y = -4 when x = 5 ⇒ (5, -4)

g'(5) = 3 means that when x = 5, the tangent slope (m) = 3

• because the first derivative gives the tangent slope

Use the Point-Slope formula to find the equation of the line that passes through the point (5, -4) with slope (m) of 3:

y - y₁ = m(x - x₁)

y -(-4) = 3(x - 5)

y + 4 = 3x - 15

y = 3x - 19