Problem 24 Graph each function. $$f(x)=\f... [FREE SOLUTION]

Chapter 12: Problem 24

Graph each function. $$f(x)=\frac{1}{4} x+2$$

Short Answer

Expert verified

To graph the function $f(x)=\frac{1}{4}x+2$, we first identify the slope ($\frac{1}{4}$) and y-intercept (2). Next, plot the y-intercept (0,2) and use the slope to find another point (4,3) on the graph. Connect these two points with a straight line to complete the graph. Verify that the graph has a slope of $\frac{1}{4}$ and a y-intercept of 2.

Step by step solution

Determine the slope and y-intercept

The given function is in the form $f(x) = mx + b$, where m is the slope and b is the y-intercept. In our case, the function is $f(x)=\frac{1}{4} x+2$, so we have: - Slope (m): $\frac{1}{4}$ - Y-intercept (b): 2

Plot the y-intercept

The y-intercept is the point where the graph intersects the y-axis. In our case, the y-intercept is at the point (0, 2). Plot this point on the graph.

Use the slope to find another point

Now we will use the slope to find another point on the graph. The slope ($\frac{1}{4}$) is the rise over the run, meaning we rise 1 unit and run 4 units: Starting from the y-intercept (0, 2), go up 1 unit (rise) and to the right 4 units (run): (0 + 4, 2 + 1) = (4, 3) So, we have another point on the graph: (4, 3). Plot this point.

Connect the two points

Now, connect the points (0, 2) and (4, 3) with a straight line. This line represents the graph of the function $f(x)=\frac{1}{4} x+2$.

Verify the graph

Check that the graph matches the key features that we determined in Step 1: - Slope: The line should go up 1 unit and to the right 4 units between points. - Y-intercept: The line should intersect the y-axis at the point (0, 2). If the graph matches these features, then we have correctly graphed the function $f(x)=\frac{1}{4} x+2$.

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Key Concepts

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

Slope and Y-Intercept

When we talk about linear functions, two important terms you will encounter are slope and y-intercept. The slope determines the steepness of the line, while the y-intercept indicates where the line crosses the y-axis. It's often expressed in the form of the equation:

The slope (m) represents the vertical change (rise) compared to the horizontal change (run). It is often written as a fraction, such as $\frac{1}{4}$, meaning the line rises 1 unit vertically for every 4 units it moves horizontally to the right.
The y-intercept (b) is where the line crosses the y-axis. In the equation $f(x) = mx + b$, this is the constant term. For the function $f(x) = \frac{1}{4}x + 2$, the y-intercept is 2.

Understanding the slope and y-intercept is crucial as it helps you quickly identify the key characteristics of a linear graph.

Plotting Points

Plotting points is a fundamental part of graphing linear functions. To plot a point, you need a pair of coordinates written as $(x, y)$. These coordinates tell you exactly where to place a point on the graph:

First, locate the x-coordinate on the x-axis (horizontal line).
Then, find the y-coordinate on the y-axis (vertical line).
Place your point where these coordinates meet.

For example, when plotting the y-intercept at (0, 2), you start at zero on the x-axis and move 2 units up the y-axis. A line through plotted points gives the visual representation of a linear function.

Linear Equations

Linear equations form straight lines when plotted on a graph, and they have the general form $y = mx + b$. This form indicates that each unit change in $x$ results in $m$ units change in $y$:

Linear functions are characterized by constant slopes.
The graph of a linear equation is a straight line.
A linear equation like $f(x) = \frac{1}{4}x + 2$ tells us that for any value of $x$, add $\frac{1}{4}$ times that value to 2 to find $y$.

This simplicity makes linear equations perfect for modeling relationships where changes happen at a constant rate. Identifying the slope and y-intercept allows you to graph a linear equation easily.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we plot points defined by pairs of numbers, known as coordinates. It consists of two perpendicular number lines:

The x-axis is horizontal.
The y-axis is vertical.
Where they intersect is the origin, marked as (0, 0).

Each point on this plane has an x-coordinate and a y-coordinate $(x, y)$. Points such as (0, 2) or (4, 3) are plotted by starting at the origin, moving x units along the x-axis, and then moving y units along the y-axis. Understanding how to use the coordinate plane is essential for graphing functions and visualizing mathematical relationships.

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91影视

Graph each function. $$f(x)=\frac{1}{4} x+2$$

Short Answer

Step by step solution

Determine the slope and y-intercept

Plot the y-intercept

Use the slope to find another point

Connect the two points

Verify the graph

Key Concepts

Slope and Y-Intercept

Plotting Points

Linear Equations

Coordinate Plane

One App. One Place for Learning.

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