Let's start by understanding the problem and solving it step-by-step.
We have two functions defined as follows:
1. [tex]\( f(x) = -\frac{3}{4} x^2 + 1 \)[/tex]
2. [tex]\( g(x) = f(x) - 3 \)[/tex]
### Part (a): Describe the transformation from the graph of [tex]\( f \)[/tex] to the graph of [tex]\( g \)[/tex]
To obtain the graph of [tex]\( g(x) \)[/tex] from the graph of [tex]\( f(x) \)[/tex], we need to analyze the transformation:
- The function [tex]\( g(x) \)[/tex] is defined as [tex]\( g(x) = f(x) - 3 \)[/tex].
- This means that [tex]\( g(x) \)[/tex] is obtained by shifting the graph of [tex]\( f(x) \)[/tex] downward by 3 units.
In simpler terms, every point on the graph of [tex]\( f(x) \)[/tex] is moved 3 units down to get the corresponding point on the graph of [tex]\( g(x) \)[/tex].
### Graph of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]
Let's set up some coordinates for both functions across a range of values for [tex]\( x \)[/tex]:
#### Table of values for [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]
[tex]\[
\begin{array}{|c|c|c|}
\hline
x & f(x) = -\frac{3}{4} x^2 + 1 & g(x) = f(x) - 3 \\
\hline
-2 & -2 & -5 \\
-1 & \frac{1}{4} & -\frac{11}{4} \\
0 & 1 & -2 \\
1 & \frac{1}{4} & -\frac{11}{4} \\
2 & -2 & -5 \\
\hline
\end{array}
\][/tex]
### Graphing [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]
Imagine plotting these values on the Cartesian coordinate plane. The graph of [tex]\( f(x) \)[/tex] is a downward opening parabola with its vertex at [tex]\( (0, 1) \)[/tex]. The graph of [tex]\( g(x) \)[/tex] is the same parabola but shifted downward by 3 units, so its vertex will be at [tex]\( (0, -2) \)[/tex].
### Part (b): Write an equation that represents [tex]\( g \)[/tex] in terms of [tex]\( x \)[/tex]
We are asked to express [tex]\( g(x) \)[/tex] purely in terms of [tex]\( x \)[/tex]. We already know:
[tex]\[ g(x) = f(x) - 3 \][/tex]
Since [tex]\( f(x) = -\frac{3}{4} x^2 + 1 \)[/tex], substitute [tex]\( f(x) \)[/tex]:
[tex]\[ g(x) = \left(-\frac{3}{4} x^2 + 1\right) - 3 \][/tex]
Simplify the equation:
[tex]\[ g(x) = -\frac{3}{4} x^2 + 1 - 3 \][/tex]
[tex]\[ g(x) = -\frac{3}{4} x^2 - 2 \][/tex]
Hence, the equation that represents [tex]\( g \)[/tex] in terms of [tex]\( x \)[/tex] is:
[tex]\[ g(x) = -\frac{3}{4} x^2 - 2 \][/tex]
In conclusion, we have described the transformation from the graph of [tex]\( f(x) \)[/tex] to [tex]\( g(x) \)[/tex] and provided the function of [tex]\( g(x) \)[/tex] in terms of [tex]\( x \)[/tex]. The graph of [tex]\( g(x) \)[/tex] is a downward parabola similar to [tex]\( f(x) \)[/tex], but shifted 3 units downward.
