Questions and Answers
Which table represents a linear 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?
y - 12 = -5(x + 2)
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?
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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.
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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)?
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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?
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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)?
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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?
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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?
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Which table represents a linear function?
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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)?
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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)?
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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
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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!
