Understanding Lines in the Plane: Equations, Slopes, and Applications in Precalculus
In this Precalculus class session, we explored key concepts related to lines in the plane, including slopes of nonvertical lines, forms of linear equations (general, point-slope, and two-point), and the conditions for parallel and perpendicular lines. Students were engaged in problem-solving exercises that involved translating real-world scenarios, such as depreciation in property value, into linear equations. We also reviewed assignments focusing on equations from prior lessons while introducing new materials that reinforce the importance of understanding lines for tackling complex functions.
Understanding Lines in the Plane: Equations, Slopes, and Applications in Precalculus
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Presentation Transcript
Precalculus Day #5 Mr. Ueland 1st Period Rm162
Today in Precalculus • Announcements • Prayer • Correct Assignment #3 [P.3, pp28-30, 1-9 odd, 21-33 odd, 37,41,49,55,66,72,73] • “New” material: “Lines in the Plane”
55.Equations and tables 55. x2 – 2x < 0 Enter as an equation using F1 View as a table using <2nd> F5 Scanning the table we notice that the only value of x that yields a negative y is x = 1
Review • Which of the following equations is equivalent to –3x < 6? • 3x < –6 • x < 10 • x > –2 • x > 2 • x > 3
Review • Which of the following equations is the solution to the equation x(x+1) = 0? • x = 0 or x = –1 • x = 0 or x = 1 • only x = –1 • only x = 0 • only x = 1
Review[What they’re working on next door] • Which of the following equations is the solution to the equation –3x ≥ 0? • x = 0 • x ≥ 0 • x ≤ 0 • x ≤ –⅓ • x is indeterminate (cannot be determined) Things get weird if you multiply both sides by zero, but not so much if you multiply zero on one side
P.3: Lines in the Plane Definitions • The slope of a nonvertical line through the points (x1, y1) and (x2, y2) is • A vertical line has no slope. • A horizontal line has zero slope “rise over run” Why? Zero in the denominator is undefined
The Many Forms of Equations of a Line • General form: • Point-slope form: A and B cannot BOTH be 0
Forms of Equations of a Line (cont). • Point-slope form: • 2-Point form:
Example 2 • Find the equation of the line with slope 2 and passing through point (–3,–4)
Parallel and Perpendicular Lines • Two nonvertical lines are parallelif and only if (iff) their slopes are equal. • Two nonvertical lines are perpendicular iff their slopesm1 and m2 are opposite reciprocals:
Example 6 • Find the equation of the line through P(2,–3) that is perpendicular to the line L with equation 4x + y = 3. Find the slope from the definition of a perp. line Use point-slope form
Example 7Applications: Everybody’s favorite! Camelot Apartments purchased a $50,000 building [either this book is old or a double-dip recession is coming] and depreciates it $2000 per year over a 25-year period. Write a linear equation giving the value y of the building in terms of the years x after the purchase. In how many years will the value of the building be $24,500?
Example 7 (cont.) This is a linear equation! Let y equal the value of the building, let the slope be the change of value (i.e. depreciation) and let b be the initial value (value today) NOTE: slope is negative. Why? That is your answer to part a)
Example 7 (cont.) We’ll use are answer to part a) to find an answer for part b), the number of years (x) when the value of the building (y) is $24,500.
Assignment 4 • P.4, pp 40-43, 1-7 odd, 11-25 odd, 37,41,45 (each part one problem), 48 • Due Monday at the start of class
