Math Problem Statement

What transformations are used in transforming the parent graph y = x to obtain the graph of c(x) = 5/9(x - 32)?

Solution

The problem asks about the transformations used to change the parent graph of y=xy = x into the graph of the temperature conversion formula c(x)=59(x32)c(x) = \frac{5}{9}(x - 32). Let's break down the transformations involved.

Given Equation:

c(x)=59(x32)c(x) = \frac{5}{9}(x - 32)

Starting with the Parent Graph:

The parent graph is the basic linear function y=xy = x, which is a straight line passing through the origin with a slope of 1.

Step-by-Step Analysis of Transformations:

  1. Horizontal Translation:

    • The formula contains (x32)(x - 32), which represents a horizontal shift to the right by 32 units. So, the graph is translated horizontally.
  2. Vertical Compression:

    • The coefficient 59\frac{5}{9} affects the slope of the line. The original slope of 1 is multiplied by 59\frac{5}{9}, which compresses the graph vertically (reducing its steepness).
  3. No Reflection in the x-axis:

    • There is no negative sign multiplying the function, so no reflection across the x-axis occurs.
  4. No Vertical Translation:

    • There is no constant added or subtracted outside the function, meaning the graph is not shifted vertically.
  5. No Reflection in the y-axis:

    • Since the xx-term remains positive, there is no reflection across the y-axis.

Conclusion:

The correct transformations are:

  • Horizontal translation
  • Vertical compression

Do you want further details or clarifications?

Here are 5 related questions:

  1. How does a horizontal translation affect the graph of a linear function?
  2. What is the effect of vertical compression on a graph’s slope?
  3. How would the graph change if the slope were negative?
  4. What are the differences between vertical and horizontal translations?
  5. How can we identify reflections in the x-axis or y-axis?

Tip:

To identify transformations, always look for changes inside the function (affecting xx) for horizontal changes and outside the function for vertical changes (affecting yy).

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Math Problem Analysis

Mathematical Concepts

Linear Transformations
Graph Transformations

Formulas

c(x) = 5/9(x - 32)
y = x

Theorems

Transformation of Functions
Horizontal and Vertical Translation
Scaling and Reflection

Suitable Grade Level

Grades 9-11