Determine whether each statement makes sense or does not make sense, and explain your reasoning. I used the ordered pairs (- 2, 2), (0, 0), and (2, 2) to graph a straight line.
Table of contents
- 0. Review of Algebra4h 18m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations1h 43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 5m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 22m
- 10. Combinatorics & Probability1h 45m
- Appendix 1. Review of Real Numbers2h 24m
- Appendix 2. Linear Equations and Inequalities3h 42m
- OLD 9. Sequences, Induction, and Probability Coming soon
- 1. - OLD - Fundamental Concepts of Algebra Coming soon
- 2. - OLD - Equations and Inequalities Coming soon
- OLD 4. Rational Functions Coming soon
- OLD 2. Functions & Graphs Coming soon
- OLD 6. Exponential and Logarithmic Functions Coming soon
- OLD 7. Systems of Equations and Inequalities Coming soon
- OLD 8. Matrices and Determinants Coming soon
- OLD 9. Conic Sections Coming soon
2. Graphs of Equations
Lines
Problem 23a
Textbook Question
Write an equation in slope-intercept form of a linear function f whose graph satisfies the given conditions. The graph of ƒ is perpendicular to the line whose equation is 3x - 2y - 4 = 0 and has the same y-intercept as this line.
Verified step by step guidance1
Rewrite the given equation 3x - 2y - 4 = 0 in slope-intercept form (y = mx + b) by isolating y. Start by moving the terms involving x and the constant to the other side: -2y = -3x + 4.
Divide through by -2 to solve for y: y = (3/2)x - 2. This gives the slope (m) of the given line as 3/2 and the y-intercept (b) as -2.
Recall that perpendicular lines have slopes that are negative reciprocals of each other. The slope of the new line will be the negative reciprocal of 3/2, which is -2/3.
Since the new line has the same y-intercept as the given line, the y-intercept (b) remains -2.
Write the equation of the new line in slope-intercept form using the slope (-2/3) and y-intercept (-2): y = (-2/3)x - 2.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Slope-Intercept Form
The slope-intercept form of a linear equation is expressed as y = mx + b, where m represents the slope and b represents the y-intercept. This form is useful for quickly identifying the slope of the line and where it crosses the y-axis, making it easier to graph linear functions.
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Slope-Intercept Form
Perpendicular Lines
Two lines are perpendicular if the product of their slopes is -1. This means that if one line has a slope of m, the other line will have a slope of -1/m. Understanding this relationship is crucial for finding the slope of a line that is perpendicular to a given line.
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Parallel & Perpendicular Lines
Finding the Y-Intercept
The y-intercept of a line is the point where the line crosses the y-axis, which occurs when x = 0. To find the y-intercept from a linear equation, you can rearrange the equation into slope-intercept form or directly substitute x = 0 into the equation. This value is essential for constructing the equation of a line with a specific y-intercept.
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Graphing Intercepts
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