Questions: Consider the following. Function Slope ------ y=x^2+8x m=-10 Determine the point(s), if any, at which the graph of the given function has a tangent line with the given slope. (If an answer does not exist, leave the space blank.)

Consider the following.

Function Slope
------
y=x^2+8x m=-10

Determine the point(s), if any, at which the graph of the given function has a tangent line with the given slope. (If an answer does not exist, leave the space blank.)

Transcript text: sider the following. \begin{tabular}{|c|c|} \hline Function & Slope \\ \hline$y=x^{2}+8 x$ & $m=-10$ \\ \hline \end{tabular} rmine the point(s), if any, at which the graph of the given function has a tangent line with the given slope. (If an answer does not \[ =(\square) \] $\square$

Solution

Solution Steps

To determine the point(s) at which the graph of the function $ y = x^2 + 8x $ has a tangent line with a given slope $ m = -10 $, we need to follow these steps:

Find the derivative of the function $ y = x^2 + 8x $ to get the slope of the tangent line at any point $ x $.
Set the derivative equal to the given slope $ m = -10 $ and solve for $ x $.
Substitute the $ x $ value(s) back into the original function to find the corresponding $ y $ value(s).

Step 1: Find the Derivative of the Function

Given the function $ y = x^2 + 8x $, we first find its derivative to determine the slope of the tangent line at any point $ x $.

\[ \frac{dy}{dx} = 2x + 8 \]

Step 2: Set the Derivative Equal to the Given Slope

We are given that the slope of the tangent line is $ m = -10 $. We set the derivative equal to this slope and solve for $ x $.

\[ 2x + 8 = -10 \]

Solving for $ x $:

\[ 2x = -18 \implies x = -9 \]

Step 3: Find the Corresponding $ y $ Value

Substitute $ x = -9 $ back into the original function to find the corresponding $ y $ value.

\[ y = (-9)^2 + 8(-9) = 81 - 72 = 9 \]