To identify the y-intercept and the zeros of the function [tex]\( g(x) = -2x^2 + 8x - 10 \)[/tex], let's analyze the function step-by-step.
### Finding the y-intercept
The y-intercept of a function occurs where [tex]\( x = 0 \)[/tex]. To find this, simply substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[
g(0) = -2(0)^2 + 8(0) - 10 = -10
\][/tex]
Thus, the y-intercept is [tex]\(-10\)[/tex].
### Finding the zeros of the function
To find the zeros of the function, we need to solve the equation [tex]\( g(x) = 0 \)[/tex]:
[tex]\[
-2x^2 + 8x - 10 = 0
\][/tex]
This is a quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex], where [tex]\( a = -2 \)[/tex], [tex]\( b = 8 \)[/tex], and [tex]\( c = -10 \)[/tex]. To find the solutions to this equation, we use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
First, compute the discriminant [tex]\( \Delta \)[/tex]:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
\Delta = 8^2 - 4(-2)(-10) = 64 - 80 = -16
\][/tex]
Since the discriminant is negative ([tex]\( \Delta < 0 \)[/tex]), there are no real zeros. Instead, there are two complex conjugate zeros. We proceed to compute these zeros using the quadratic formula:
[tex]\[
x = \frac{-8 \pm \sqrt{-16}}{2(-2)} = \frac{-8 \pm 4i}{-4} = 2 \pm i
\][/tex]
Thus, the zeros of the function are [tex]\( 2 - i \)[/tex] and [tex]\( 2 + i \)[/tex].
### Summary
1. The y-intercept of the function [tex]\( g(x) = -2x^2 + 8x - 10 \)[/tex] is [tex]\(-10\)[/tex].
2. The zeros of the function are the complex numbers [tex]\( 2 - i \)[/tex] and [tex]\( 2 + i \)[/tex].
