Question

for the quadratic function $f(x)=x^2 + 2x - 8$, answer parts (a) through (c)

(a) find the vertex and the axis of symmetry of the quadratic function, and determine whether the graph is concave up or concave down.
the vertex is \\(\square\\).
(type an ordered pair, using integers or fractions.)
what is the equation of the axis of symmetry?
the axis of symmetry is \\(\square\\).
(use integers or fractions for any numbers in the equation.)
is the graph concave up or concave down?
\\(\bigcirc\\) concave down
\\(\bigcirc\\) concave up (selected)

(b) find the y - intercept and the x - intercepts, if any.
what is the y - intercept? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
\\(\bigcirc\\) a. the y - intercept is \\(\square\\).
(type an integer or a simplified fraction.)
\\(\bigcirc\\) b. there is no y - intercept.
what is the x - intercept? select the correct choice below and, if necessary, fill in the answer box to complete your choice.

Explanation:

Step1: Find the vertex of the quadratic function

For a quadratic function in the form \( f(x) = ax^2 + bx + c \), the x - coordinate of the vertex is given by \( x = -\frac{b}{2a} \). In the function \( f(x)=x^{2}+2x - 8 \), \( a = 1 \), \( b = 2 \), and \( c=-8 \).
So, \( x=-\frac{2}{2\times1}=- 1 \).
To find the y - coordinate, substitute \( x = - 1 \) into the function: \( f(-1)=(-1)^{2}+2\times(-1)-8=1 - 2 - 8=-9 \).
So the vertex is \( (-1,-9) \).

Step2: Find the axis of symmetry

The axis of symmetry of a quadratic function \( f(x)=ax^{2}+bx + c \) is the vertical line \( x = -\frac{b}{2a} \). We already found that \( x=-\frac{2}{2\times1}=-1 \), so the equation of the axis of symmetry is \( x=-1 \).

Step3: Determine concavity

For a quadratic function \( f(x)=ax^{2}+bx + c \), if \( a>0 \), the graph is concave up; if \( a < 0 \), the graph is concave down. In \( f(x)=x^{2}+2x - 8 \), \( a = 1>0 \), so the graph is concave up.

Step4: Find the y - intercept

The y - intercept of a function \( y = f(x) \) is found by setting \( x = 0 \). Substitute \( x = 0 \) into \( f(x)=x^{2}+2x - 8 \): \( f(0)=0^{2}+2\times0 - 8=-8 \). So the y - intercept is \( - 8 \) (option A: The y - intercept is \(-8\)).

Step5: Find the x - intercepts

To find the x - intercepts, set \( f(x)=0 \), so we solve the equation \( x^{2}+2x - 8 = 0 \).
Factor the quadratic equation: \( x^{2}+2x - 8=(x + 4)(x - 2)=0 \).
Set each factor equal to zero: \( x+4 = 0 \) or \( x - 2=0 \).
Solving \( x+4 = 0 \) gives \( x=-4 \), and solving \( x - 2=0 \) gives \( x = 2 \). So the x - intercepts are \( x=-4 \) and \( x = 2 \).

Answer:

(a) The vertex is \((-1,-9)\); The axis of symmetry is \(x = - 1\); The graph is concave up.
(b) The y - intercept is \(-8\) (A. The y - intercept is \(-8\)); The x - intercepts are \(x=-4\) and \(x = 2\).