Questions and Answers
What is the equation of the line that passes through the points (-3,5) and (0,2)?
y = -1x + 2
What is the equation for the linear function f that satisfies f(3) = 5 and f(0) = -7?
y = 4x - 7
What is the equation in slope-intercept form of the line that passes through the points (3,2) and (4,9)?
y = 7x - 19
What is the equation in slope-intercept form of the linear function f that with the values f(-4) = -8 and f(4) = 4?
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What is the slope and y-intercept of the linear function represented by y = -2/3x + 7?
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What is the equation in point-slope form of the line that passes through (5, -2) and (9, 6)?
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What is the equation in point-slope form of the linear function f that satisfies f(-2) = 3 and f(4) = 9?
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What is the graph of the equation y - 2 = 3/2(x + 5)?
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Write two additional equations that are equivalent to 8x + 4y = 6.
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What are the equations of the horizontal and vertical lines that pass through the point (3, -7)?
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What is the equation in standard form of the line that passes through (1,5) and (5,-3)?
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Find the value of B so that the line 2x + By = -4 passes through the point (-3,7). What is B?
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What is the equation of a line passing through the point (2, -3) that is parallel to y = 2/3x - 5?
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What is the equation of a line passing through the point (2, -3) that is perpendicular to y = -4/5x + 8?
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What do you need to do to draw a line of best fit?
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Find the equation in slope-intercept form of the line of best fit drawn in question 16, using points (10,100) and (18,400).
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Describe the correlation going from left to right.
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Describe the correlation going from right to left.
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Describe the correlation where points are everywhere.
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Find the zero of f(x) = 5x - 7.
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Use your calculator to find the line of best fit for the following data: X = 10, 15, 33, 50, 55, 60; Y = 27, 16, 53, 76, 88, 91. What do A and B equal?
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Study Notes
Writing Linear Equations
- To find the equation of a line given two points, calculate the slope ( m ) and y-intercept ( b ). Example: from points (-3, 5) and (0, 2), ( m = -1 ) and ( b = 2 ), resulting in ( y = -1x + 2 ).
- For a linear function given specific inputs: If ( f(3) = 5 ) and ( f(0) = -7 ), slope ( m = 4 ) and y-intercept ( b = -7 ) lead to ( y = 4x - 7 ).
Slope-Intercept Form
- Use point coordinates to derive an equation in slope-intercept form: From (3, 2) and (4, 9), slope ( m = 7 ) leads to the equation ( y = 7x - 19 ).
- Given points (−4, -8) and (4, 4), with slope ( m = 3/2 ), results in ( y = \frac{3}{2}x - 2 ).
Graphing and Points-Slope Form
- For the equation ( y = -\frac{2}{3}x + 7 ), the slope ( m = -\frac{2}{3} ) and y-intercept is 7, which determines the graph.
- From points (5, -2) and (9, 6), in point-slope form with ( m = 2 ): ( y + 2 = 2(x - 5) ).
Equivalent Equations
- An equation can be rewritten in multiple forms. ( 8x + 4y = 6 ) can be converted to ( 16x + 8y = 12 ) or ( 4x + 2y = 3 ).
Horizontal and Vertical Lines
- The equation for vertical lines through (3, -7) is ( x = 3 ), while the horizontal line is ( y = -7 ).
Standard Form and Finding B
- Standard form can be generated from two points: using (1,5) and (5,-3), derive ( 2x + y = 7 ).
- To ensure a line passes through a specific point, such as (-3, 7) for the equation ( 2x + By = -4 ), solve for ( b = \frac{2}{7} ).
Parallel and Perpendicular Lines
- Lines can be constructed based on characteristics of other lines: For parallel line through (2, -3) to ( y = \frac{2}{3}x - 5 ), use slope ( m = \frac{2}{3} ).
- For perpendicular lines, such as one from (2, -3) to ( y = -\frac{4}{5}x + 8 ), use the negative reciprocal slope ( m = \frac{5}{4} ).
Correlation and Line of Best Fit
- The correlation of a line can vary: Positive when moving left to right, negative when right to left, or no correlation when points are scattered.
- A line of best fit involves calculating slope and intercept from data points. For instance, from points (10, 100) and (18, 400), ( m = 37.5 ) and ( b = -275 ).
Finding Zeros
- Zeros in functions, such as for ( f(x) = 5x - 7 ), can be found where the function equals zero; here, ( x = \frac{7}{5} ).
Linear Regression Output
- Linear regression can determine the best-fit line for given data sets, producing coefficients ( A ) and ( B ). In one case, given arrays produce ( A = 1.47 ) and ( B = 3.79 ).
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Description
This quiz focuses on the principles of writing linear equations, including finding slope and y-intercept from given points. You'll explore slope-intercept and point-slope forms, as well as the concepts of equivalent equations. Test your understanding of graphing and representing linear functions with various methods.
