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2.3 (Review) Linear Functions and Slope In this section we will review: using an equation to graph a line, and using information about a line to write its equation. Our focus will be on graphs and applications. (See p229#92.) Do answers to module.
Definition of Slope (m) The slope of the line through the distinct points (x1, y1) and (x2, y2) is where x2 – x1≠ 0. y Run x2 – x1 Change in y Rise y2 – y1 = = y2 Change in x Run x2 – x1 Rise y2 – y1 (x2, y2) y1 (x1, y1) x x1 x2 Definition of Slope
Zero Slope Positive Slope y y m > 0 m = 0 x x Negative Slope Undefined Slope y y Line rises from left to right. Line is horizontal. m is undefined m < 0 x x Line falls from left to right. Line is vertical. The Possibilities for a Line’s Slope
The point-slope equation of a nonvertical line of slope m that passes through the point (x1, y1) is y – y1 = m(x – x1). Point-Slope Form of the Equation of a Line
The slope-intercept equation of a nonvertical line with slope m and y-intercept b is y = mx + b We can easily write this in function notation: f(x) =mx + b Slope-Intercept Form of the Equation of a Line
Graphing y=mx+b Using the Slope and y-Intercept • Graphing y = mx + b by Using the Slope and y-Intercept • Plot the y-intercept on the y-axis. This is the point (0, b)… • Obtain a second point using the slope, m. Write m as a fraction, and use rise over run starting at the y-intercept to plot this point. • Use a straightedge to draw a line through the two points. Draw arrowheads at the ends of the line to show that the line continues indefinitely in both directions. Simply put: “ANCHOR” at the point (y-intercept), move in the direction of the slope, label the coordinates, connect the dots..
y = - 2/3x + 2 The slope is - 2/3. The y-intercept is 2. Text Example Graph the line whose equation is y = - 2/3 x + 2. SolutionThe equation of the line is in the form y = mx + b. We can find the slope, m, by identifying the coefficient of x. We can find the y-intercept, b, by identifying the constant term.
Graph the line whose equation is y = - 2/3x + 2. Text Example cont. • Solution: • “ANCHOR” at ________ • MOVE _________________ or ________________. • 3. LABEL at least two points. Q: How would this process differ for a slope of say, 2/3?
2x – 3y = - 6 Text Example Find the slope and the y-intercept of the line whose equation is 2x – 3y = - 6 . Solution Once in the proper form, the coefficient of ______, _________, is the slope and the constant term, ______________, is the __________-intercept.
A horizontal line is given by an equation of the form y = b where b is the y-intercept. Note: An equation whose graph is a horizontal line may not be solved for y, but there will be no “x” in the equation. You must solve for y yourself. Ex: given 2y-8=5, you would get y= 13/2, a horizontal line going through 6.5 on the y-axis. Equation of a Horizontal Line
A vertical line is given by an equation of the form x = a where a is the x-intercept. Note: An equation whose graph is a vertical line may not be solved for x, but there will be no “y” in the equation. You must solve for x yourself. Ex: given –3x+7=4, you would get x= 1, a vertical line going through 1 on the x-axis. Equation of a Vertical Line
Every line has an equation that can be written in the general form Ax + By + C = 0 Where A, B, and C are three integers (not fractions or decimals, or radicals if possible), A and B are not both zero, and the leading coefficient is positive. General Form of the Equation of a Line
Point-slope form: y – y1 = m(x – x1) Slope-intercept form: y = mx + b Horizontal line: y = b Vertical line: x = a General form: Ax + By + C = 0 Standard form: Ax + By = C Slope (m) = Equations of Lines
Writing the Equation of a Line given information about the line: • Must always know the SLOPE of the line, plus one other piece of information. • Given slope & y- intercept, use SLOPE-INTERCEPT form y = mx +b • Given slope and point, or two points, use POINT-SLOPE form y – y1 = m(x – x1). • (Section 2.4) Given point and a parallel or perpendicular line, “use” the line to find the slope and use POINT-SLOPE form. • “General form” Ax + By + C = 0 is just “pretty”.
. Write the equation of the line in both point slope and slope intercept form which passes through (-1,3) with a slope of – 5/3. Text Example Solution
Slope and Parallel Lines 2.4 More on Slope • If two nonvertical lines are parallel, then they have the same slope. • If two distinct nonvertical lines have the same slope, and different y-intercepts, then they are parallel. • Two distinct vertical lines, both with undefined slopes, are parallel.
Solution We are looking for the equation of the line shown on the left. Notice that the line passes through the point (-3, 2). Using the _____________________________ form of the line’s equation, we have x1 = -3 and y1 = 2. y – y1 = m(x – x1) 5 y = 2x + 1 4 (-3, 2) 3 Run = 1 2 1 Rise = 2 y1 = 2 x1 = -3 -5 -4 -3 -2 -1 1 2 3 4 5 -1 -2 -3 -4 -5 Example: Write an equation of the line passing through (-3, 2) and parallel to the line whose equation is y = 2x + 1. Express the equation in y-intercept form.
y – y1 = m(x – x1) 5 y = 2x + 1 4 (-3, 2) 3 Run = 1 2 1 Rise = 2 y1 = 2 m= 2 x1 = -3 -5 -4 -3 -2 -1 1 2 3 4 5 -1 -2 -3 -4 -5 Example cont. Write an equation of the line passing through (-3, 2) and parallel to the line whose equation is y = 2x + 1. Express the equation in y-intercept form. Solution Parallel lines have the same slope. Because the slope of the given line is______, m = _____ for the new equation. Note: we only “use” the line y = 2x +1 to get the slope, after that we do not need it anymore.
y – 2 = 2x + 6 Apply the distributive property. y = 2x + 8 Add 2 to both sides. This is the slope-intercept form of the equation. Example cont. Write an equation of the line passing through (-3, 2) and parallel to the line whose equation is y = 2x + 1. Express the equation in point-slope form and y-intercept form. Solution The point-slope form of the line’s equation is y – 2 = 2[x – (-3)] y – 2 = 2(x + 3) Solving for y, we obtain the slope-intercept form of the equation.
90° Slope and Perpendicular Lines Two lines that intersect at a right angle (90°) are said to be perpendicular. There is a relationship between the slopes of perpendicular lines. • Slope and Perpendicular Lines • If two nonvertical lines are perpendicular, then the product of their slopes is –1. • If the product of the slopes of two lines is –1, then the lines are perpendicular. • A horizontal line having zero slope is perpendicular to a vertical line having undefined slope. • We will simply find the NEGATIVE RECIPROCAL to write the slope of a perpendicular line. Ex: If line one has a slope of 7 and line two is perpendicular to line one, then the slope of line 2 will be –1/7.
2x + 4y – 4 = 0 This is the given equation. Find the slope of any line that is perpendicular to the line whose equation is 2x + 4y – 4 = 0. Example Solution We begin by writing the equation of the given line in _______________________________ form. The given line has slope ___________. Any line perpendicular to this line has a slope that is the negative reciprocal, ________. Do p 227 # 58, p 238 # 20 (graph each)
The Average Rate of Change of a Function (See p 235 height as a function of age & discuss.) Let be distinct points on the graph of a function f. The average rate of change of f from x1 to x2 is:
Ex: Calculate the average rate of change of the function: What do you observe from your result?