Problem 24

Question

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

Step-by-Step Solution

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Answer

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.

1Step 1: 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

2Step 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.

3Step 3: 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.

4Step 4: 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 $f(x)=\frac{1}{4} x+2$.

5Step 5: 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 $f(x)=\frac{1}{4} x+2$.

Key Concepts

Slope and Y-InterceptPlotting PointsLinear EquationsCoordinate Plane

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.

Problem 24

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