Question
Instructions: Use algebraic techniques to solve problems, unless otherwise instructed. Show the calculations leading to your answers (except basic arithmetic), when there is work to show. Simplify and circle your final answers, 1. (18 pts.) A function \( f \) is given and the indicated transformations are to be applied to its graph in the given order. Write the equation (i.e, the function rule) for the new function with the transformed graph. (a) \( f(x)=\frac{1}{x} \), reflect in the \( x \) axis, shift up by 3
Ask by Little Higgins. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The transformed function is \( f(x) = -\frac{1}{x} + 3 \).
Solution
We begin with the original function:
f(x) = 1/x
Step 1. Reflect the graph across the x-axis.
A reflection across the x-axis changes f(x) to –f(x). Thus,
–f(x) = –(1/x) = –1/x
Step 2. Shift the graph upward by 3 units.
A vertical shift upward by 3 adds 3 to the function. Therefore,
–1/x + 3
Final Answer (circled):
f(x) = –1/x + 3
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Simplify this solution Beyond the Answer
To find the new function after the indicated transformations to the graph of \( f(x) = \frac{1}{x} \), we'll apply the transformations step by step. 1. **Reflect in the x-axis**: Reflecting the function in the x-axis changes the sign of the function. Thus, we have: \[ g(x) = -f(x) = -\frac{1}{x} \] 2. **Shift up by 3**: Shifting a function upwards by \( 3 \) units adds \( 3 \) to the entire function. Hence, we have: \[ h(x) = g(x) + 3 = -\frac{1}{x} + 3 \] Now, combining these transformations, the equation of the new function is: \[ h(x) = -\frac{1}{x} + 3 \] We can box our final answer: \[ \boxed{h(x) = -\frac{1}{x} + 3} \]
