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To find the point \((x, y)\) at which the graph of \( y = 5x^2 + 2x - 6 \) has a horizontal tangent line, we need to follow these steps:

### 1. Compute the derivative of the function:
The slope of the tangent line at any point is given by the derivative of the function \( y = 5x^2 + 2x - 6 \). Let's differentiate it with respect to \(x\).

\[
y' = \frac{d}{dx}(5x^2 + 2x - 6)
\]

The derivative is:
\[
y' = 10x + 2
\]

### 2. Set the derivative equal to zero:
A horizontal tangent line occurs where the slope is zero, i.e., when \( y' = 0 \).

\[
10x + 2 = 0
\]

Solve for \(x\):

\[
10x = -2
\]
\[
x = -\frac{1}{5}
\]

### 3. Find the corresponding \(y\)-coordinate:
Substitute \( x = -\frac{1}{5} \) into the original function \( y = 5x^2 + 2x - 6 \) to find the corresponding \(y\)-coordinate.

\[
y = 5\left(-\frac{1}{5}\right)^2 + 2\left(-\frac{1}{5}\right) - 6
\]
\[
y = 5\left(\frac{1}{25}\right) + 2\left(-\frac{1}{5}\right) - 6
\]
\[
y = \frac{5}{25} - \frac{2}{5} - 6
\]
\[
y = \frac{1}{5} - \frac{2}{5} - 6
\]
\[
y = -\frac{1}{5} - 6
\]
\[
y = -\frac{1}{5} - \frac{30}{5} = -\frac{31}{5}
\]

### Final Answer:
The point at which the graph has a horizontal tangent line is \( \left(-\frac{1}{5}, -\frac{31}{5}\right) \).

Would you like further clarification or details?

### Related Questions:
1. How do you find the derivative of higher-degree polynomials?
2. What are the conditions for a tangent line to be horizontal?
3. Can a function have more than one point with a horizontal tangent line?
4. How does the second derivative help determine the nature of the horizontal tangent point?
5. How would the solution change if the function had a cubic term?

### Tip:
Always check your work by substituting the \(x\)-value back into the derivative to confirm that the slope is zero.

Find the point (x,y) at which the graph of y = 5x^2 + 2x - 6 has a horizontal tangent line.

Explore the concept of horizontal tangent lines in calculus with a detailed solution to finding the point where y = 5x^2 + 2x - 6 has a horizontal tangent. This guide covers differentiation of quadratic functions, solving equations to find zeros of derivatives, and substituting back to find the y-value.

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