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Questions and Answers
Which equation represents a linear function?
Which equation represents a linear function?
What is the slope of the line represented by the equation $y = 2x + 5$?
What is the slope of the line represented by the equation $y = 2x + 5$?
Which of the following points lies on the line represented by the equation $y = 4x - 1$?
Which of the following points lies on the line represented by the equation $y = 4x - 1$?
What is the y-intercept of the line given by the equation $y = -3x + 6$?
What is the y-intercept of the line given by the equation $y = -3x + 6$?
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What is the general form of a linear function?
What is the general form of a linear function?
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If a linear function has a slope of $-2$ and passes through the point $(3, 4)$, what is the equation of the line?
If a linear function has a slope of $-2$ and passes through the point $(3, 4)$, what is the equation of the line?
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What is the slope of the line represented by the equation $3x - 4y = 12$ when rewritten in slope-intercept form?
What is the slope of the line represented by the equation $3x - 4y = 12$ when rewritten in slope-intercept form?
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A linear function is represented by the equation $y = mx + 7$. If the value of $m$ is halved, what is the new y-intercept if the original slope is $4$?
A linear function is represented by the equation $y = mx + 7$. If the value of $m$ is halved, what is the new y-intercept if the original slope is $4$?
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If the function $f(x) = 5x - 3$ is translated 2 units up, what is the new equation?
If the function $f(x) = 5x - 3$ is translated 2 units up, what is the new equation?
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Which of the following scenarios describes a linear function correctly?
Which of the following scenarios describes a linear function correctly?
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Study Notes
- A linear function is a function whose graph is a straight line. It can be represented by the equation y = mx + b, where 'm' is the slope and 'b' is the y-intercept.
Key Characteristics of Linear Functions
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The slope (m) represents the rate of change of y with respect to x. A positive slope indicates an upward trend, a negative slope a downward trend, and a zero slope a horizontal line.
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The y-intercept (b) is the point where the line crosses the y-axis. It represents the value of y when x is zero.
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The equation is of the first degree (the highest power of the variable is 1).
Graphing Linear Functions
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To graph a linear function, two points are sufficient. Find two points that satisfy the equation (x, y) and plot them on a coordinate plane. Draw a straight line passing through these two points.
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Alternatively, you can use the slope-intercept form (y = mx + b) directly: plot the y-intercept (0, b), then use the slope (m) to find a second point. Rise over run (m = rise/run) determines the change in y for a given change in x.
Finding the Equation of a Line
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Given two points (x₁, y₁) and (x₂, y₂), the slope can be calculated as m = (y₂ - y₁)/(x₂ - x₁)
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After finding the slope, use the point-slope form, y - y₁ = m(x - x₁), to find the equation. Substitute one of the points and the slope into this form.
Special Cases of Linear Functions
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Horizontal lines: These have a slope of zero, and their equation is of the form y = b.
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Vertical lines: These have an undefined slope (division by zero) and their equation is of the form x = a.
Applications of Linear Functions
- Many real-world situations involve linear relationships. Examples include:
- Calculating total cost (if there's a constant rate like cost per item).
- Determining distance traveled if speed is constant.
- Modeling growth or decay at a fixed rate.
Determining the Slope and Y-Intercept
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Given an equation in the form y = mx + b: the slope is 'm', and the y-intercept is 'b'.
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Sometimes an equation is given in a different form and needs to be rearranged to the slope-intercept form. For example, if you are given 2x + 3y = 6, rearrange to isolate 'y'.
Example Problems (Illustrative)
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Find the slope and y-intercept of the line described by the equation 2x - 4y = 8.
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Rearrange the equation to slope-intercept form (y = mx + b):
- 2x - 8 = 4y
- y = (1/2)x -2
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The slope is 1/2 and the y-intercept is -2.
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Find the equation of the line passing through the points (3, 1) and (-1, 5).
- Calculate the slope: m = (5 - 1) / (-1 - 3) = -1.
- Use the point-slope form: y - 1 = -1(x - 3)
- Simplify to get the equation: y = -x + 4
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Description
Explore the key characteristics of linear functions, including slope and y-intercept. Learn how to graph these functions using two points and the slope-intercept form. Test your understanding of the concepts applicable to linear equations.
