Let's analyze the given quadratic function [tex]\( f(x) = \frac{1}{5} x^2 - 5 x + 12 \)[/tex] and determine the truth of the given statements.
1. The value of [tex]\( f(-10) = 82 \)[/tex]:
To find [tex]\( f(-10) \)[/tex]:
[tex]\[
f(-10) = \frac{1}{5}(-10)^2 - 5(-10) + 12
\][/tex]
[tex]\[
f(-10) = \frac{1}{5}(100) + 50 + 12
\][/tex]
[tex]\[
f(-10) = 20 + 50 + 12 = 82
\][/tex]
Therefore, it is true that [tex]\( f(-10) = 82 \)[/tex].
2. The graph of the function is a parabola:
The given function is a quadratic function of the form [tex]\( ax^2 + bx + c \)[/tex]. Since it is quadratic, its graph is indeed a parabola. Thus, this statement is true.
3. The graph of the function opens down:
The coefficient of [tex]\( x^2 \)[/tex] (the term [tex]\(\frac{1}{5}\)[/tex]) determines the direction in which the parabola opens.
Since [tex]\(\frac{1}{5}\)[/tex] is positive, the parabola opens upwards, not downwards. Therefore, this statement is false.
4. The graph contains the point [tex]\( (20, -8) \)[/tex]:
To check if the graph contains the point [tex]\((20, -8)\)[/tex]:
[tex]\[
f(20) = \frac{1}{5}(20)^2 - 5(20) + 12
\][/tex]
[tex]\[
f(20) = \frac{1}{5}(400) - 100 + 12
\][/tex]
[tex]\[
f(20) = 80 - 100 + 12 = -8
\][/tex]
Therefore, it is true that the point [tex]\((20, -8)\)[/tex] lies on the graph.
5. The graph contains the point [tex]\( (0, 0) \)[/tex]:
To check if the graph contains the point [tex]\((0, 0)\)[/tex]:
[tex]\[
f(0) = \frac{1}{5}(0)^2 - 5(0) + 12
\][/tex]
[tex]\[
f(0) = 12
\][/tex]
[tex]\( f(0) = 12 \)[/tex], not 0. So, the point [tex]\((0, 0)\)[/tex] does not lie on the graph. Therefore, this statement is false.
So, the three true statements about the function and its graph are:
- The value of [tex]\( f(-10) = 82 \)[/tex].
- The graph of the function is a parabola.
- The graph contains the point [tex]\( (20, -8) \)[/tex].
