Introduction
In unit 3 of linear functions, students learn about the fundamental concepts of linear equations, including slope, intercepts, and graphing. This article will provide an answer key for unit 3 linear functions, which will serve as a valuable resource for students and teachers alike. By having access to the answer key, students can check their work, identify any mistakes, and gain a better understanding of the concepts covered in unit 3. Let's dive into the answer key for unit 3 linear functions!
1. Slope
1.1 Definition
Slope is a measure of how steep a line is. It represents the rate of change between two points on a line. The formula for finding the slope between two points (x1, y1) and (x2, y2) is given by:
Slope = (y2 - y1) / (x2 - x1)
1.2 Examples
Example 1: Find the slope between the points (2, 4) and (6, 10).
Slope = (10 - 4) / (6 - 2) = 6 / 4 = 3/2
Example 2: Find the slope between the points (-3, 5) and (1, -1).
Slope = (-1 - 5) / (1 - (-3)) = -6 / 4 = -3/2
2. Intercepts
2.1 x-intercept
The x-intercept is the point where the line intersects the x-axis. To find the x-intercept, set y = 0 and solve for x. The x-intercept is represented as (x, 0).
2.2 y-intercept
The y-intercept is the point where the line intersects the y-axis. To find the y-intercept, set x = 0 and solve for y. The y-intercept is represented as (0, y).
2.3 Examples
Example 1: Find the x-intercept and y-intercept of the line with equation y = 2x + 3.
For x-intercept, set y = 0:
0 = 2x + 3
-3 = 2x
x = -3/2
Therefore, the x-intercept is (-3/2, 0).
For y-intercept, set x = 0:
y = 2(0) + 3
y = 3
Therefore, the y-intercept is (0, 3).
3. Graphing Linear Equations
3.1 Slope-Intercept Form
The slope-intercept form of a linear equation is given by y = mx + b, where m represents the slope and b represents the y-intercept. This form makes it easy to graph a linear equation.
3.2 Examples
Example 1: Graph the line with equation y = 2x + 3.
Plot the y-intercept (0, 3).
Use the slope to find additional points:
- Starting from the y-intercept (0, 3), move 1 unit to the right and 2 units up to get the point (1, 5).
- Starting from the point (1, 5), move 1 unit to the right and 2 units up to get the point (2, 7).
- Starting from the point (1, 5), move 1 unit to the left and 2 units down to get the point (-1, 1).
Connect the points to form a line.
4. Writing Linear Equations
4.1 Point-Slope Form
The point-slope form of a linear equation is given by y - y1 = m(x - x1), where m represents the slope and (x1, y1) represents a point on the line. This form is useful for writing an equation given a point and the slope.
4.2 Examples
Example 1: Write the equation of a line with slope 3 that passes through the point (2, 4).
Using the point-slope form:
y - 4 = 3(x - 2)
y - 4 = 3x - 6
y = 3x - 2
Therefore, the equation of the line is y = 3x - 2.
5. Systems of Linear Equations
5.1 Solving by Graphing
To solve a system of linear equations by graphing, plot the lines on the same graph and find the point of intersection.
5.2 Solving by Substitution
To solve a system of linear equations by substitution, solve one equation for one variable and substitute it into the other equation. Solve the resulting equation for the remaining variable.
5.3 Solving by Elimination
To solve a system of linear equations by elimination, add or subtract the equations to eliminate one variable. Solve the resulting equation for the remaining variable.
5.4 Examples
Example 1: Solve the system of equations using the graphing method:
y = 2x + 1
y = -x + 4
Graph the lines y = 2x + 1 and y = -x + 4.
Find the point of intersection, which is (1, 3).
Therefore, the solution to the system of equations is x = 1 and y = 3.
Conclusion
The answer key provided in this article serves as a valuable resource for students and teachers studying unit 3 linear functions. By having access to the answer key, students can check their work, identify any mistakes, and gain a better understanding of the concepts covered in unit 3. Understanding the concepts of slope, intercepts, graphing linear equations, and solving systems of linear equations is crucial for success in algebra and beyond. With this answer key, students can confidently tackle problems related to linear functions.