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.
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Vertex: Given as \( (4, 2) \).
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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 \( (6, 0) \).
- 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: \( (0, -6) \)
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