Determine the coordinates of the vertex.

The vertex is the highest, or in this case, the lowest point on the parabola.

First, find the *x*-coordinate of the vertex using the following formula.

*x* = −

*b*

2*a*

From the given equation

*y* = 5*x*2 + 20*x* + 6,

you know *a* = 5 and *b* = 20. Substitute these values into the equation for determining the *x*-coordinate of the vertex.

*x*

=

−

*b*

2*a*

=

−

(20)

2

=

−

20

=

Next, to find the *y*-coordinate of the vertex, substitute the value of *x* found above into the original equation. Then, simplify using the correct order of operations.

*y*

=

5*x*2 + 20*x* + 6

=

5

2

+ 20   + 6

=

5   − 40 + 6

=

− 40 + 6

=

Therefore, the vertex is located at what point?

(*x*, *y*) =

Let's go step by step to find the coordinates of the vertex for the given quadratic equation.

The quadratic equation is:

\[
y = 5x^2 + 20x + 6
\]

### Step 1: Find the *x*-coordinate of the vertex

The formula for the *x*-coordinate of the vertex is:

\[
x = \frac{-b}{2a}
\]

From the equation \(y = 5x^2 + 20x + 6\), we know that:
- \(a = 5\)
- \(b = 20\)

Now, substitute these values into the formula:

\[
x = \frac{-20}{2(5)} = \frac{-20}{10} = -2
\]

So, the *x*-coordinate of the vertex is \(-2\).

### Step 2: Find the *y*-coordinate of the vertex

Now substitute \(x = -2\) into the original equation to find the corresponding *y*-coordinate:

\[
y = 5(-2)^2 + 20(-2) + 6
\]

Simplifying:

\[
y = 5(4) + 20(-2) + 6
\]
\[
y = 20 - 40 + 6
\]
\[
y = -20 + 6
\]
\[
y = -14
\]

### Final Answer:
The vertex is located at the point \((-2, -14)\).

---

Would you like further details or have any questions?

Here are five related questions for deeper understanding:
1. How does changing the value of \(a\) in a quadratic equation affect the position of the vertex?
2. What is the geometric significance of the vertex in the context of a parabola?
3. How can you determine if the vertex represents the highest or lowest point on the parabola?
4. How would you find the axis of symmetry for this quadratic equation?
5. What happens to the vertex if you modify the constant term in the quadratic equation?

**Tip:** The sign of \(a\) in a quadratic equation \(y = ax^2 + bx + c\) determines whether the parabola opens upwards (\(a > 0\)) or downwards (\(a < 0\)).

Determine the coordinates of the vertex. The vertex is the highest, or in this case, the lowest point on the parabola. First, find the *x*-coordinate of the vertex using the following formula. *x* = − *b* / 2 *a*. From the given equation *y* = 5*x*^2 + 20*x* + 6, you know *a* = 5 and *b* = 20. Substitute these values into the equation for determining the *x*-coordinate of the vertex. Next, to find the *y*-coordinate of the vertex, substitute the value of *x* found above into the original equation.

Learn how to determine the coordinates of the vertex for the quadratic equation y = 5x^2 + 20x + 6. This step-by-step guide shows you how to use the vertex formula to find the x- and y-coordinates. Suitable for students in Grades 8-10.

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