Podcast
Questions and Answers
Which table represents a linear function?
Which table represents a linear function?
Which equation represents the graphed function?
Which equation represents the graphed function?
y = 1/4x - 2
What is the point-slope form of the equation of the line Mr. Shaw graphed?
What is the point-slope form of the equation of the line Mr. Shaw graphed?
y - 12 = -5(x + 2)
Which linear function represents the line given by the point-slope equation y + 7 = -(x + 6)?
Which linear function represents the line given by the point-slope equation y + 7 = -(x + 6)?
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Which linear function is represented by the graph?
Which linear function is represented by the graph?
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How can Kendra determine if the function is actually linear?
How can Kendra determine if the function is actually linear?
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Which equations also represent the line that passes through the points (-4, 10) and (-1, 5)? Check all that apply.
Which equations also represent the line that passes through the points (-4, 10) and (-1, 5)? Check all that apply.
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What must m be for the function to be linear, according to Mrs. Jackson's table?
What must m be for the function to be linear, according to Mrs. Jackson's table?
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If the equation of the line is written in slope-intercept form, y = mx + b, what is the value of b for line MN passing through points M(4, 3) and N(7, 12)?
If the equation of the line is written in slope-intercept form, y = mx + b, what is the value of b for line MN passing through points M(4, 3) and N(7, 12)?
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Which linear function represents the line given by the point-slope equation y - 8 = 1/2 (x - 4)?
Which linear function represents the line given by the point-slope equation y - 8 = 1/2 (x - 4)?
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How does the graph of g(x) compare to the graph of f(x) when Timmy writes the equation f(x) = x - 1 and creates g(x) = x - 2?
How does the graph of g(x) compare to the graph of f(x) when Timmy writes the equation f(x) = x - 1 and creates g(x) = x - 2?
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What is the slope of the line represented by the equation 2x + 5y = 10?
What is the slope of the line represented by the equation 2x + 5y = 10?
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Which equation represents line QR that goes through points Q(0, 1) and R(2, 7)?
Which equation represents line QR that goes through points Q(0, 1) and R(2, 7)?
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Which equation represents the direct variation function that contains the points (-9, -3) and (-12, -4)?
Which equation represents the direct variation function that contains the points (-9, -3) and (-12, -4)?
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Does the line drawn through (-4, 3) and (4, 3) represent a direct variation?
Does the line drawn through (-4, 3) and (4, 3) represent a direct variation?
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What is the slope of the linear function represented in the table?
What is the slope of the linear function represented in the table?
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Which equation correctly uses the point (-2, -6) to write the equation of this line in point-slope form?
Which equation correctly uses the point (-2, -6) to write the equation of this line in point-slope form?
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Which table represents a linear function?
Which table represents a linear function?
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What is the slope of the function represented in the table?
What is the slope of the function represented in the table?
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How does the slope of g(x) compare to the slope of f(x)?
How does the slope of g(x) compare to the slope of f(x)?
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Which linear function represents the line given by the point-slope equation y - 8 = 1/2 (x - 4)?
Which linear function represents the line given by the point-slope equation y - 8 = 1/2 (x - 4)?
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What is the constant of variation, k, of the line y = kx through (3,18) and (5,30)?
What is the constant of variation, k, of the line y = kx through (3,18) and (5,30)?
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Study Notes
Linear Functions and Relationships
- A linear function can be identified by observing consistent rates of change between sets of ordered pairs. For example, checking if the change in y (output) over the change in x (input) remains constant across pairs.
- The slope-intercept form of a linear equation is represented as y = mx + b, where m indicates the slope and b is the y-intercept.
Point-Slope Form
- The point-slope form of a line can be derived from a specific point that lies on the line and its slope, expressed as y - y₁ = m(x - x₁).
Equations and Slope
- The equation f(x) = -5x + 2 indicates a linear function with a slope of -5 and a y-intercept of 2.
- A function shown in both point-slope and slope-intercept forms conveys the same linear relationship.
Characteristics of Graphs
- A linear graph through points (M: 4, 3; N: 7, 12) can be transformed to identify the y-intercept (b), which equals -9 in slope-intercept form.
- A graph that is horizontal or vertical does not represent a direct variation unless it passes through the origin.
Rates of Change
- To identify whether a function is linear, compare the rate of change between various ordered pairs:
- Consistent rates indicate linearity.
- Discrepancies reveal nonlinear behavior.
Variations and Constraints
- A direct variation requires that the relationship can be expressed as y = kx, with k being the constant of variation, reflecting a proportionality.
- Nonlinear relationships may show alterations in the rate of change as the graphs curve or show varying slopes.
Function Analysis
- The slope of a function can be calculated through differences in y-values over differences in x-values.
- Functions can be analyzed through tables or graphs to determine their slopes and overall linear representations.
Examples and Applications
- When given points like (-4, 10) and (-1, 5), their linear representation can also be expressed in multiple equivalent forms.
- A direct relation can be modeled through functions: for instance, y = x/3 captures the direct variation of given points.
Conclusion
- Understanding linear relationships involves mastering the concepts of slopes, point-slope equations, and the ability to determine linearity through various mathematical forms and data manipulation.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your understanding of linear and exponential relationships with this unit test. Included are various questions about identifying linear functions, writing equations, and working with point-slope forms. Challenge yourself with these flashcards to reinforce your learning!
