Question

Identify these key features of the parabola. (2 points) vertex: (4,2) (smaller) x-intercept: (2,0) (larger) x-intercept: () y-intercept: ()

Answer 1

To identify the key features of the parabola given its vertex and one x-intercept, we can use the vertex form of a parabola equation and the properties of parabolas.

1. **Vertex**: Given as \( (4, 2) \).

2. **X-intercepts**: One x-intercept is given as \( (2, 0) \). To find the other x-intercept, we can use the fact that a parabola is symmetric about its vertex. The vertex's x-coordinate is at \( 4 \) and one x-intercept is at \( 2 \).

The distance from the vertex ( \( x = 4 \) ) to the x-intercept ( \( x = 2 \) ) is \( 4 - 2 = 2 \). By symmetry, there will be another x-intercept located the same distance from the vertex on the other side. Thus:

\[
\text{Other x-intercept} = 4 + 2 = 6
\]

So the other x-intercept is \( (6, 0) \).

3. **Y-intercept**: To find the y-intercept, we need to evaluate the parabola at \( x = 0 \).

Given the vertex form \( y = a(x - h)^2 + k \) where \( (h, k) \) is the vertex:

\[
y = a(x - 4)^2 + 2
\]

We can use one of the x-intercepts to find the value of \( a \). Let's use the intercept at \( (2, 0) \):

Substituting \( x = 2 \) and \( y = 0 \):

\[
0 = a(2 - 4)^2 + 2 \\
0 = a(2)^2 + 2 \\
0 = 4a + 2 \\
4a = -2 \\
a = -\frac{1}{2}
\]

Now we have the equation of the parabola:

\[
y = -\frac{1}{2}(x - 4)^2 + 2
\]

Now we can find the y-intercept by substituting \( x = 0 \):

\[
y = -\frac{1}{2}(0 - 4)^2 + 2 \\
y = -\frac{1}{2}(16) + 2 \\
y = -8 + 2 \\
y = -6
\]

So the y-intercept is \( (0, -6) \).

### Summary of Key Features:
- Vertex: \( (4, 2) \)
- X-intercepts: \( (2, 0) \) and \( (6, 0) \)
- Y-intercept: \( (0, -6) \)