Determine the coordinates of the point on the graph of  f(x) = - 5x2 - 2x + 10 where the slope of the graph is equal to -1.    

To find the coordinates of the point on the graph of \( f(x) = -5x^2 - 2x + 10 \) where the slope of the graph is equal to \(-1\), we need to follow these steps:

1. **Find the derivative of \( f(x) \)**: The derivative \( f'(x) \) gives us the slope of the graph at any point \( x \).
2. **Set the derivative equal to \(-1\)**: This will allow us to find the \( x \)-coordinate where the slope is \(-1\).
3. **Find the corresponding \( y \)-coordinate**: Substitute the \( x \)-value back into the original function \( f(x) \) to find the \( y \)-coordinate.

Let's go through these steps.

### 1. Find the derivative of \( f(x) \):

\[
f(x) = -5x^2 - 2x + 10
\]

Taking the derivative:

\[
f'(x) = \frac{d}{dx} \left(-5x^2 - 2x + 10\right) = -10x - 2
\]

### 2. Set the derivative equal to \(-1\):

\[
f'(x) = -10x - 2 = -1
\]

Solve for \( x \):

\[
-10x = -1 + 2 = 1 \implies x = -\frac{1}{10}
\]

### 3. Find the corresponding \( y \)-coordinate:

Substitute \( x = -\frac{1}{10} \) back into the original function \( f(x) \):

\[
f\left(-\frac{1}{10}\right) = -5\left(-\frac{1}{10}\right)^2 - 2\left(-\frac{1}{10}\right) + 10
\]

Calculate the value:

\[
f\left(-\frac{1}{10}\right) = -5\left(\frac{1}{100}\right) + \frac{2}{10} + 10 = -\frac{5}{100} + \frac{2}{10} + 10 = -\frac{1}{20} + \frac{1}{5} + 10
\]

\[
f\left(-\frac{1}{10}\right) = -0.05 + 0.2 + 10 = 10.15
\]

### Final Answer:

The coordinates of the point on the graph where the slope is equal to \(-1\) are:

\[
\left(-\frac{1}{10}, 10.15\right) \approx (-0.1, 10.15)
\]

Would you like more details or have any questions? Here are five related questions:

1. How do you find the derivative of a polynomial function?
2. What is the significance of setting the derivative equal to a specific value?
3. How can the second derivative help determine the concavity of a function at a point?
4. How would the coordinates change if the slope was set to a different value?
5. How does changing the coefficients of the function affect the slope at any given point?

**Tip:** When finding the slope of a function at a specific point, always start by calculating the derivative, as it represents the slope of the tangent to the curve at any given \( x \)-value.

Discover how to find the coordinates of a point on the graph of f(x) = -5x^2 - 2x + 10 where the slope is equal to -1. Step-by-step solution including differentiation, solving for x, and substituting to find y. Suitable for advanced high school students and calculus learners.

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