Get started for free

Log In Start studying!

Get started for free Log out

Chapter 2: Problem 25

Suppose the point \((8,12)\) is on the graph of \(y=f(x) .\) Find a point on thegraph of each function. (a) \(y=f(x+4)\) (b) \(y=f(x)+4\)

### Short Answer

Expert verified

(a) (4,12) (b) (8,16)

## Step by step solution

01

## Identify the transformation for part (a)

For the function transformation given by \(y=f(x+4)\), the term \((x+4)\) indicates a horizontal shift. Specifically, it shifts the graph of \(f(x)\) to the left by 4 units.

02

## Apply the transformation for part (a)

To find the new coordinates after the horizontal shift, subtract 4 from the original x-coordinate of the point (8,12). The new x-coordinate is \(8-4=4\). The y-coordinate remains the same, so the new point is \( (4,12) \).

03

## Identify the transformation for part (b)

For the function transformation given by \(y=f(x)+4\), the term \(+4\) indicates a vertical shift. Specifically, it shifts the graph of \(f(x)\) upwards by 4 units.

04

## Apply the transformation for part (b)

To find the new coordinates after the vertical shift, add 4 to the original y-coordinate of the point (8,12). The new y-coordinate is \(12+4=16\). The x-coordinate remains the same, so the new point is \( (8,16) \).

## Key Concepts

These are the key concepts you need to understand to accurately answer the question.

###### Horizontal Shift

Horizontal shifts involve moving the entire graph of a function left or right along the x-axis. This translation does not alter the shape of the graph, but rather its position relative to the y-axis.

When we have an equation of the form \(y=f(x+c)\), it indicates a horizontal shift:

- If c is positive, there is a leftward shift by c units.
- If c is negative, the shift is rightward by the absolute value of c units.

In our exercise, we started with the point (8,12) on the graph of \(y=f(x)\). For the transformation \(y=f(x+4)\), the graph undergoes a leftward shift by 4 units. Therefore, the x-coordinate of the point moves from 8 to 4, with the y-coordinate remaining unchanged. The new point on the graph for \(y=f(x+4)\) is \( (4,12) \).

###### Vertical Shift

Vertical shifts involve moving the entire graph of a function up or down along the y-axis. Similar to horizontal shifts, this transformation does not change the shape of the graph, merely its position relative to the x-axis.

When we look at an equation like \(y=f(x)+c\), it signals a vertical shift:

- If c is positive, the graph shifts upwards by c units.
- If c is negative, the graph shifts downwards by the absolute value of c units.

Referring to our exercise again, we have the point (8,12) on the graph of \(y=f(x)\). For the transformation \(y=f(x)+4\), the graph is shifted upwards by 4 units. Hence, the y-coordinate of the point increases from 12 to 16, while the x-coordinate remains the same. The new point on the graph for \(y=f(x)+4\) is \( (8,16) \).

###### Graphing Functions

Graphing functions involves plotting points on a coordinate plane that satisfy the function's equation to reveal the overall shape and position of the graph.

Understanding various transformations is key to predicting how these modifications will affect the graph:

**Horizontal Shifts:**Move the graph left or right.**Vertical Shifts:**Move the graph up or down.**Reflections:**Flip the graph over a particular axis.**Stretching/Shrinking:**Alter the graph's width or height.

When you graph a function and apply transformations step-by-step, it's easier to understand these changes. For example, starting with a known point makes it straightforward to see the resulting position after shifts.

By using the exercise's initial point (8,12) and applying a horizontal shift to get (4,12) and a vertical shift to get (8,16), you can visualize how the graph moves in the coordinate plane with these transformations.

### One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

## Most popular questions from this chapter

## Recommended explanations on Math Textbooks

### Applied Mathematics

Read Explanation### Mechanics Maths

Read Explanation### Logic and Functions

Read Explanation### Calculus

Read Explanation### Pure Maths

Read Explanation### Decision Maths

Read ExplanationWhat do you think about this solution?

We value your feedback to improve our textbook solutions.

## Study anywhere. Anytime. Across all devices.

Sign-up for free

This website uses cookies to improve your experience. We'll assume you're ok with this, but you can opt-out if you wish. Accept

Privacy & Cookies Policy

#### Privacy Overview

This website uses cookies to improve your experience while you navigate through the website. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. We also use third-party cookies that help us analyze and understand how you use this website. These cookies will be stored in your browser only with your consent. You also have the option to opt-out of these cookies. But opting out of some of these cookies may affect your browsing experience.

Always Enabled

Necessary cookies are absolutely essential for the website to function properly. This category only includes cookies that ensures basic functionalities and security features of the website. These cookies do not store any personal information.

Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. It is mandatory to procure user consent prior to running these cookies on your website.