Question

for the quadratic function ( f(x)=x^{2}+4x - 12 ), answer parts (a) through (f).
is the graph concave up or concave down?
concave down
concave up
(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.
a. the y - intercept is ( - 12 ).
(type an integer or a simplified fraction.)
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.
a. the x - intercept(s) is/are
(type an integer or a simplified fraction. use a comma to separate answers as needed.)
b. there is/are no x - intercept(s).

Explanation:

Step1: Find the y - intercept

For a quadratic function \(y = f(x)=ax^{2}+bx + c\), the y - intercept is found by setting \(x = 0\).
Substitute \(x = 0\) into \(f(x)=x^{2}+4x - 12\):
\(f(0)=0^{2}+4\times0 - 12=- 12\)

Step2: Find the x - intercepts

Set \(y = f(x)=0\), so we have the quadratic equation \(x^{2}+4x - 12 = 0\).
We can solve it using the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) for the quadratic equation \(ax^{2}+bx + c = 0\). Here \(a = 1\), \(b = 4\), \(c=-12\).
First, calculate the discriminant \(\Delta=b^{2}-4ac=(4)^{2}-4\times1\times(-12)=16 + 48=64\).
Then \(x=\frac{-4\pm\sqrt{64}}{2\times1}=\frac{-4\pm8}{2}\).
For the plus - sign: \(x=\frac{-4 + 8}{2}=\frac{4}{2}=2\).
For the minus - sign: \(x=\frac{-4-8}{2}=\frac{-12}{2}=-6\).

Answer:

A. The y - intercept is \(-12\).
A. The x - intercept(s) is/are \(2,-6\).