Sure! Let's go through the process of solving the given quadratic function [tex]\( f(x) = 3x^2 - 18x + 24 \)[/tex].
### a) The [tex]\( y \)[/tex]-intercept:
The [tex]\( y \)[/tex]-intercept of a function occurs where the graph intersects the [tex]\( y \)[/tex]-axis. This happens when [tex]\( x = 0 \)[/tex].
To find it, we substitute [tex]\( x = 0 \)[/tex] into the quadratic function:
[tex]\[ f(0) = 3(0)^2 - 18(0) + 24 \][/tex]
[tex]\[ f(0) = 24 \][/tex]
So, the [tex]\( y \)[/tex]-intercept is:
[tex]\[ (0, 24) \][/tex]
### b) The [tex]\( x \)[/tex]-intercepts:
The [tex]\( x \)[/tex]-intercepts are the points where the function intersects the [tex]\( x \)[/tex]-axis. This occurs when [tex]\( f(x) = 0 \)[/tex]. Thus, we solve the equation:
[tex]\[ 3x^2 - 18x + 24 = 0 \][/tex]
To solve this quadratic equation, we can use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\( a = 3 \)[/tex], [tex]\( b = -18 \)[/tex], and [tex]\( c = 24 \)[/tex].
First, we need to find the discriminant ([tex]\( \Delta \)[/tex]):
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = (-18)^2 - 4(3)(24) \][/tex]
[tex]\[ \Delta = 324 - 288 \][/tex]
[tex]\[ \Delta = 36 \][/tex]
Since the discriminant is positive, we have two distinct real roots.
Next, we find the roots using the quadratic formula:
[tex]\[ x = \frac{-(-18) \pm \sqrt{36}}{2(3)} \][/tex]
[tex]\[ x = \frac{18 \pm 6}{6} \][/tex]
This gives us two solutions:
[tex]\[ x_1 = \frac{18 + 6}{6} = \frac{24}{6} = 4 \][/tex]
[tex]\[ x_2 = \frac{18 - 6}{6} = \frac{12}{6} = 2 \][/tex]
So, the [tex]\( x \)[/tex]-intercepts are:
[tex]\[ (4, 0) \text{ and } (2, 0) \][/tex]
### Final results:
a) The [tex]\( y \)[/tex]-intercept is [tex]\( (0, 24) \)[/tex].
b) The [tex]\( x \)[/tex]-intercepts are [tex]\( (4, 0) \)[/tex] and [tex]\( (2, 0) \)[/tex].
