Use the accompanying graph of y=f(x) to sketch a graph of equation. (a) y=f(x-1)-2 (b) y=-f(x)+1

step 1 So in this problem we're given a graph of y equals f of x, and we're asked to sketch a graph of

Use the accompanying graph of y=f(x) to sketch a graph of...

Key Concepts

Function Transformations

This concept deals with altering the graph of a function through different operations such as translations, reflections, and scalings. Function transformations allow one to predict and sketch a new graph based on the algebraic modifications made to the original function, providing a systematic approach to understanding how changes in the formula affect the function’s appearance.

Horizontal Translation

A horizontal translation shifts the graph of a function left or right. When the function is written as f(x - h), it results in a shift to the right by h units, and when written as f(x + h), it shifts the graph to the left by h units. This concept is central when adjusting input values to translate the graph along the x-axis without altering its shape.

Vertical Translation

Vertical translation involves moving the graph of a function upward or downward. This is accomplished by adding or subtracting a constant from the function, such as in f(x) + k or f(x) - k, which shifts the graph upward or downward by k units respectively. It changes the y-values while preserving the general shape of the graph.

Vertical Reflection

Vertical reflection flips the graph of a function over the x-axis. This transformation is indicated by multiplying the function by -1, changing the sign of all output values. It is useful for analyzing how the graph behaves with respect to the x-axis and provides a mirror image of the original graph in the vertical direction.

Horizontal Scaling

Horizontal scaling modifies the width of the graph by stretching or compressing it along the x-axis. When the variable x is multiplied by a factor or replaced by a fraction, such as f(bx) with b < 1 or b > 1, the graph is stretched or compressed respectively. Understanding horizontal scaling is essential for predicting how changes in the input will affect the spread of the graph.

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