Math  /  Algebra

QuestionThe graphs of y=f(x),y=g(x)y=f(x), y=g(x) and y=h(x)y=h(x) are shown below. Remember that y=g(x)y=g(x) and y=h(x)y=h(x) are transformations of y=f(x)y=f(x). Write equations for g(x)g(x) and h(x)h(x) in terms of f(x)f(x). a. b.

Studdy Solution

STEP 1

What is this asking? We need to describe how the graphs of g(x)g(x) and h(x)h(x) are related to f(x)f(x) using equations. Watch out! Don't mix up reflections and shifts!
A reflection flips the graph, while a shift moves it.

STEP 2

1. Analyze g(x)g(x)
2. Analyze h(x)h(x)

STEP 3

Look closely!
The graph of g(x)g(x) looks like f(x)f(x) flipped upside down.
This tells us g(x)g(x) is a **reflection** of f(x)f(x) across the x-axis.

STEP 4

A reflection across the x-axis means we multiply the *output* of the function by 1-1.
So, g(x)=1f(x)g(x) = -1 \cdot f(x), which simplifies to g(x)=f(x)g(x) = -f(x).

STEP 5

The graph of h(x)h(x) looks like f(x)f(x) but shifted **upwards**.
How far up?
It looks like it's moved 2 units\textbf{2 units} up!
This is a **vertical shift**.

STEP 6

A vertical shift upwards by 2 units means we **add 2** to the output of f(x)f(x).
So, h(x)=f(x)+2h(x) = f(x) + 2.

STEP 7

g(x)=f(x)g(x) = -f(x) h(x)=f(x)+2h(x) = f(x) + 2

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