Answer:
the equation of the tangent line to the graph of f(x) at the point (-4, 9) is y = -8x - 23.
Explanation:
To find the slope of the function's graph at the point (-4, 9), you can calculate the derivative of the function f(x) = x² - 7. The derivative will give you the slope of the tangent line at any point on the graph.
f(x) = x² - 7
Now, find the derivative of f(x):
f'(x) = 2x
Now, let's find the slope at the point (-4, 9):
f'(-4) = 2 * (-4) = -8
So, the slope of the graph at the point (-4, 9) is -8.
Now, you can use the point-slope form of a line to find the equation of the tangent line. The point-slope form is:
y - y₁ = m(x - x₁),
where (x₁, y₁) is the point (-4, 9) and m is the slope (-8):
y - 9 = -8(x - (-4))
y - 9 = -8(x + 4)
Now, you can simplify this to the slope-intercept form (y = mx + b):
y - 9 = -8x - 32
Add 9 to both sides:
y = -8x - 32 + 9
y = -8x - 23
So, the equation of the tangent line to the graph of f(x) at the point (-4, 9) is y = -8x - 23.