Problem 64 Graph the linear function and la... [FREE SOLUTION]

Chapter 2: Problem 64

Graph the linear function and label the $x$ -intercept. $$f(x)=x+3$$

Short Answer

Expert verified

Graph the line through (0, 3) and (-3, 0), with x-intercept at (-3, 0).

Step by step solution

Identify the y-intercept

The linear function is given as $f(x) = x + 3$. This equation is in the slope-intercept form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. For this function, the y-intercept $b$ is 3. This means the graph will cross the y-axis at (0, 3).

Identify the slope

In the slope-intercept form $y = mx + b$, the coefficient $m$, which is 1 in this case, represents the slope of the line. A slope of 1 means for every unit increase in $x$, $y$ increases by 1 unit.

Find the x-intercept

The x-intercept is found by setting $f(x) = 0$ and solving for $x$. This gives $0 = x + 3$. Solving for $x$, we subtract 3 from both sides to get $x = -3$. So, the x-intercept is at (-3, 0).

Plot key points and draw the line

Now, plot the y-intercept (0, 3) on the graph. Also, plot the x-intercept (-3, 0). Use the slope of 1 to add another point by moving 1 unit right and 1 unit up from the y-intercept, reaching (1, 4). Draw a straight line through these points to represent the function.

Label the x-intercept on the graph

On the graph you've drawn, clearly label the x-intercept at (-3, 0). This point is where the line crosses the x-axis and represents the solution to $x + 3 = 0$.

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Key Concepts

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

Graphing Linear Equations

Graphing a linear equation involves plotting points on a coordinate plane that represent the solutions to the equation. These points line up to form a straight line. Linear equations can be expressed in the form $y = mx + b$, known as the slope-intercept form. Here, $m$ is the slope, and $b$ is the y-intercept. To graph a linear equation, identify these components:

The y-intercept, where the graph crosses the y-axis, is one of the key points.
The x-intercept, found by setting $y = 0$ and solving for $x$, is another crucial point.
Plot these intercepts on the cartesian plane to help visualize the line.

Once these points are plotted, select one or more additional points using the slope, and then draw a line through these points. Extending in both directions, the line will depict all the solutions of the equation.

x-intercept

The x-intercept of a linear equation is a point where the graph crosses the x-axis. At this point, the value of $y$ is zero.
To find the x-intercept, set the equation equal to zero and solve for $x$. For the equation $f(x) = x + 3$, solve:

Set $f(x) = 0$: $0 = x + 3$.
Solve for $x$ by subtracting 3 from both sides: $x = -3$.
The x-intercept is $(-3, 0)$.

This x-intercept can be plotted on the graph. The point $(-3, 0)$ is where the line touches the x-axis. Graphically, it marks the solution to the equation $x + 3 = 0$. By correctly identifying the x-intercept, you'll understand where the line crosses horizontally.

Slope

A slope is a crucial aspect of a linear equation. It represents the 'steepness' and the direction of the line.
In the slope-intercept form $y = mx + b$, $m$ stands for the slope. It shows how much $y$ changes when $x$ increases by one unit. If $m = 1$, like in our function $f(x) = x + 3$, it means:

For every increase of 1 in $x$, $y$ increases by 1.
The line rises diagonally upward from left to right at an angle of 45 degrees.
The change is consistent across all points on the line.

Understanding slope helps in predicting the behavior of linear equations and drawing the line accurately. Once you have the slope, you can start from any point on the line and find others by moving according to the rise over run of the slope.

Slope-Intercept Form

The slope-intercept form is one way to express a linear equation graphically and algebraically. Expressed as $y = mx + b$, it provides a clear view of the line's properties:

$m$ is the slope: It dictates the direction and angle of the line on the graph.
$b$ is the y-intercept: This is where the line crosses the y-axis, at point $(0, b)$.

This form helps in graphing because it readily identifies the y-intercept, where you can start your graph. From this point, the slope tells you how to proceed:

Increase $x$ values to move along the line using the slope.
The slope helps in finding additional points for more accurate drawing.

Slope-intercept form is versatile and widely used due to its straightforward breakdown of linear equations, making graphing simpler and more intuitive.

91影视

Graph the linear function and label the \(x\) -intercept. $$f(x)=x+3$$

Short Answer

Step by step solution

Identify the y-intercept

Identify the slope

Find the x-intercept

Plot key points and draw the line

Label the x-intercept on the graph

Key Concepts

Graphing Linear Equations

x-intercept

Slope

Slope-Intercept Form

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