Consider the quadratic equation
y
=
−
2
x
2
+
32
.

Determine the following: (enter all numerical answers as integers or decimals. Round your answer to two decimal places as needed.):

The smallest
x
-intercept is
x
=
.
The largest
x
-intercept is
x
=
.
The
y
-intercept is
y
=
.
The vertex is ( ,  ).
The line of symmetry has the equation  .

The quadratic equation given is:

\[
y = -2x^2 + 32
\]

### 1. **Find the x-intercepts:**

To find the \(x\)-intercepts, set \(y = 0\) and solve for \(x\):

\[
0 = -2x^2 + 32
\]

\[
2x^2 = 32
\]

\[
x^2 = 16
\]

\[
x = \pm 4
\]

Thus, the smallest \(x\)-intercept is \(x = -4\) and the largest \(x\)-intercept is \(x = 4\).

### 2. **Find the y-intercept:**

The \(y\)-intercept occurs when \(x = 0\):

\[
y = -2(0)^2 + 32 = 32
\]

Thus, the \(y\)-intercept is \(y = 32\).

### 3. **Find the vertex:**

The vertex of a parabola in the form \(y = ax^2 + bx + c\) is given by:

\[
x_{\text{vertex}} = -\frac{b}{2a}
\]

Here, \(a = -2\) and \(b = 0\) (there is no \(x\) term), so:

\[
x_{\text{vertex}} = -\frac{0}{2(-2)} = 0
\]

To find the \(y\)-coordinate of the vertex, substitute \(x = 0\) into the original equation:

\[
y_{\text{vertex}} = -2(0)^2 + 32 = 32
\]

Thus, the vertex is \((0, 32)\).

### 4. **Find the line of symmetry:**

The line of symmetry for a parabola is a vertical line that passes through the vertex. Since the vertex is at \(x = 0\), the line of symmetry is:

\[
x = 0
\]

### Summary of Results:

- The smallest \(x\)-intercept is \(x = -4\).
- The largest \(x\)-intercept is \(x = 4\).
- The \(y\)-intercept is \(y = 32\).
- The vertex is \((0, 32)\).
- The line of symmetry is \(x = 0\).

Would you like more details or have any questions?

Here are 5 follow-up questions:
1. How does the coefficient of \(x^2\) affect the shape of the parabola?
2. What happens to the \(x\)-intercepts if the constant term is negative?
3. How can you determine the direction in which the parabola opens?
4. Can a quadratic equation have no real \(x\)-intercepts?
5. How does the line of symmetry relate to the vertex?

**Tip:** The line of symmetry always passes through the vertex of a parabola.

Consider the quadratic equation y = −2x^2 + 32. Determine the following: The smallest x-intercept is x = . The largest x-intercept is x = . The y-intercept is y = . The vertex is ( , ). The line of symmetry has the equation .

Explore how to solve the quadratic equation y = −2x^2 + 32, find x and y intercepts, the vertex, and the line of symmetry. Suitable for high school students in Grades 9-12.

Math Problem Statement

Consider the quadratic equation y

The smallest x -intercept is x

. The largest x -intercept is x

. The y -intercept is y

Solution

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