Unit 3 Graphing Linear Functions
Vocabulary
| Key Terms | Meaning | Description | Picture |
|--------------------------------------------|-----------------------------------------|-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|------------------------------|
| U3 L1 Linear Equation | An equation that models a line | A linear equation can be expressed in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. | |
| U3 L2 Ratio & Proportion | A relationship between quantities | A ratio is a comparison of two quantities, while proportion shows that two ratios are equal. |
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| U3 L3 Proportional Relationship | A constant ratio between two variables | A proportional relationship can be expressed as \( y = kx \), where \( k \) is the constant of proportionality. |
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| U3 L3 Unit Rate from a table | Rate per one unit | A unit rate is the ratio of two measurements in which the second quantity is one. For example, if a table shows that 4 apples cost $2, the unit rate is $2/4 apples = $0.50 per apple. |
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| U3 L4 Graphing a proportional relationship from a Linear equation | Visual representation of proportion | When graphed, a proportional relationship is a straight line that passes through the origin (0, 0) with a slope equal to the constant of proportionality. |
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| U3 L4 Slope as rate of change | Measure of steepness | Slope represents the rate of change between the variables \( y \) and \( x \) in a linear function. A positive slope indicates an increasing relationship, while a negative slope indicates a decreasing one. |
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| U3 L5 Slope as a unit rate | Slope representing rate per unit | In a proportional relationship, the slope can be interpreted as the unit rate, which is the value of \( y \) when \( x = 1 \). |
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| U3 L5 Slope using Similar Triangles | Use of triangles to find slope | By creating a right triangle between two points on a line, the slope can be calculated as the ratio of the rise (change in y) to the run (change in x). |
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| U3 L6 Comparing proportional relationships in tables, graphs, and equations | Analysis of relationships | Changing slope affects steepness; a higher slope indicates a steeper line. Changing the y-intercept changes where the line crosses the y-axis. |
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| U3 L7 Linear Equation in Slope-Intercept Form | Standard form for linear equations | The slope-intercept form is \( y = mx + b \). The y-intercept (\( b \)) is the point where the line crosses the y-axis at \( (0, b) \). |
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| U3 L8 Change a Linear equation into Slope-Intercept form | Converting forms | Convert equations from standard form \( Ax + By = C \) to slope-intercept form \( y = mx + b \) or from point-slope form \( y - y_1 = m(x - x_1) \) to slope-intercept form. |
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| U3 L9 Graphing Linear Equations | Steps to graph equations | 1. Get to slope-intercept form. 2. Graph the y-intercept. 3. Use the slope to plot another point. 4. Connect the points with a line. |
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| U3 L10 Show y = mx + b as a translation of y = mx | Understanding equation transformations | When a constant \( b \) is added, the line shifts up or down without changing its slope, representing a vertical translation. |
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Word Problem Analysis
Business Model Example:
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Proportional Equation:
- The service charges $3 per hour. The equation representing this proportional relationship is:
\[ y = 3x \]
where \( y \) is the total cost and \( x \) is the number of hours.
- The service charges $3 per hour. The equation representing this proportional relationship is:
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New Equation with Flat Rate:
- With a flat rate of $10 plus $3 per hour, the new equation becomes:
\[ y = 3x + 10 \]
where \( y \) is the total cost, \( x \) is the number of hours, and \( 10 \) represents the flat fee.
- With a flat rate of $10 plus $3 per hour, the new equation becomes:
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Graphical Comparison:
- Both equations yield straight lines, with the first line passing through the origin (0, 0) and having a slope of 3. The second line has the same slope but a y-intercept of 10.
- Similarities: Both equations represent linear relationships.
- Differences: The first line starts at the origin, while the second starts at (0, 10). The second line is always 10 units above the first.
This outline covers key terms related to graphing linear functions, along with their meanings and graphical representations. The word problem ties the concepts together, illustrating how they can be applied in real-world scenarios.
9 answers