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Find the [tex]\( y \)[/tex]- and [tex]\( x \)[/tex]-intercepts of the quadratic function [tex]\( f(x) = x^2 + 2x - 8 \)[/tex].

Select the correct answer below:

A. [tex]\( y \)[/tex]-intercept: [tex]\((0, -8)\)[/tex], [tex]\( x \)[/tex]-intercepts: [tex]\((-2, 0)\)[/tex] and [tex]\((4, 0)\)[/tex].

B. [tex]\( y \)[/tex]-intercept: [tex]\((0, 8)\)[/tex], [tex]\( x \)[/tex]-intercepts: [tex]\((2, 0)\)[/tex] and [tex]\((-4, 0)\)[/tex].

C. [tex]\( y \)[/tex]-intercept: [tex]\((0, -8)\)[/tex], [tex]\( x \)[/tex]-intercepts: [tex]\((2, 0)\)[/tex] and [tex]\((-4, 0)\)[/tex].

D. [tex]\( y \)[/tex]-intercept: [tex]\((0, 8)\)[/tex], [tex]\( x \)[/tex]-intercepts: [tex]\((-2, 0)\)[/tex] and [tex]\((4, 0)\)[/tex].



Answer :

GinnyAnswer
To find the [tex]\(y\)[/tex]- and [tex]\(x\)[/tex]-intercepts of the quadratic function [tex]\(f(x) = x^2 + 2x - 8\)[/tex], we'll follow specific steps.

### Finding the [tex]\(y\)[/tex]-intercept

The [tex]\(y\)[/tex]-intercept is the point where the graph of the function crosses the [tex]\(y\)[/tex]-axis. This occurs when [tex]\(x = 0\)[/tex].

1. Substitute [tex]\(x = 0\)[/tex] in the function:
[tex]\[ f(0) = 0^2 + 2 \cdot 0 - 8 = -8 \][/tex]

So, the [tex]\(y\)[/tex]-intercept is [tex]\((0, -8)\)[/tex].

### Finding the [tex]\(x\)[/tex]-intercepts

The [tex]\(x\)[/tex]-intercepts are the points where the graph of the function crosses the [tex]\(x\)[/tex]-axis. This occurs when [tex]\(f(x) = 0\)[/tex].

1. Set the function equal to zero:
[tex]\[ x^2 + 2x - 8 = 0 \][/tex]

2. Solve the quadratic equation [tex]\(x^2 + 2x - 8 = 0\)[/tex]. We can factorize this quadratic equation.

We look for two numbers that multiply to [tex]\(-8\)[/tex] and add up to [tex]\(2\)[/tex]. These numbers are [tex]\(4\)[/tex] and [tex]\(-2\)[/tex].
[tex]\[ x^2 + 2x - 8 = (x + 4)(x - 2) = 0 \][/tex]

3. Set each factor equal to zero to find the [tex]\(x\)[/tex]-values at the intercepts:
[tex]\[ x + 4 = 0 \quad \text{or} \quad x - 2 = 0 \][/tex]
[tex]\[ x = -4 \quad \text{or} \quad x = 2 \][/tex]

So, the [tex]\(x\)[/tex]-intercepts are [tex]\((-4, 0)\)[/tex] and [tex]\((2, 0)\)[/tex].

### Conclusion

The correct answer is:
- [tex]\(y\)[/tex]-intercept: [tex]\((0, -8)\)[/tex]
- [tex]\(x\)[/tex]-intercepts: [tex]\((2, 0)\)[/tex] and [tex]\((-4, 0)\)[/tex]

Therefore, the correct option is:
[tex]\[ \text{y-intercept: } (0, -8) . \text{x-intercepts: } (2, 0) \text{ and } (-4, 0). \][/tex]

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