117 Views

Download Presentation
##### 2.3 (Review) Linear Functions and Slope

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**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?