(a) To write the quadratic function in vertex form, we can complete the square. This involves rearranging the quadratic expression into a perfect square trinomial plus a constant.
(b) Once the function is in vertex form, the vertex can be directly identified from the expression as it will be in the form \( a(x-h)^2 + k \), where \((h, k)\) is the vertex.
(c) To find the \(x\)-intercepts, set the function equal to zero and solve for \(x\). This can be done using the quadratic formula or by factoring if possible.
The given function is \( p(x) = 2x^2 - 8x - 1 \). To convert this quadratic function into vertex form, we need to complete the square.
Factor out the coefficient of \( x^2 \) from the first two terms:
\[
p(x) = 2(x^2 - 4x) - 1
\]
Complete the square inside the parentheses:
- Take half of the coefficient of \( x \), square it, and add and subtract it inside the parentheses.
- The coefficient of \( x \) is \(-4\). Half of \(-4\) is \(-2\), and \((-2)^2 = 4\).
Add and subtract 4 inside the parentheses:
\[
p(x) = 2(x^2 - 4x + 4 - 4) - 1
\]
Simplify by completing the square:
\[
p(x) = 2((x - 2)^2 - 4) - 1
\]
Distribute the 2 and simplify:
\[
p(x) = 2(x - 2)^2 - 8 - 1 = 2(x - 2)^2 - 9
\]
Thus, the vertex form of the function is:
\[
p(x) = 2(x - 2)^2 - 9
\]
The vertex form of a quadratic function is \( a(x - h)^2 + k \), where \((h, k)\) is the vertex.
From the vertex form \( p(x) = 2(x - 2)^2 - 9 \), we can see that:
Therefore, the vertex is \((2, -9)\).
To find the \(x\)-intercepts, set \( p(x) = 0 \) and solve for \( x \):
\[
2(x - 2)^2 - 9 = 0
\]
Add 9 to both sides:
\[
2(x - 2)^2 = 9
\]
Divide by 2:
\[
(x - 2)^2 = \frac{9}{2}
\]
Take the square root of both sides:
\[
x - 2 = \pm \sqrt{\frac{9}{2}}
\]
Solve for \( x \):
\[
x = 2 \pm \sqrt{\frac{9}{2}}
\]
Simplify:
\[
x = 2 \pm \frac{3\sqrt{2}}{2}
\]
Thus, the \(x\)-intercepts are:
\[
x = 2 + \frac{3\sqrt{2}}{2} \quad \text{and} \quad x = 2 - \frac{3\sqrt{2}}{2}
\]
- (a) Vertex form: \(\boxed{p(x) = 2(x - 2)^2 - 9}\)
- (b) Vertex: \(\boxed{(2, -9)}\)
- (c) \(x\)-Intercepts: \(\boxed{x = 2 \pm \frac{3\sqrt{2}}{2}}\)
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