Unit 6 Linear Equations and their Graphs - PowerPoint
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Unit 6: Writing Linear Equations Warm up Retail Sales A music store is offering a coupon promotion on its CDs. The regular price for CDs is $14.With the coupon, customers are given $4 off the total purchase. The equation t = 14c - 4, where c is the number of CDs and t is the total cost of the purchase, models this situation. a.) Graph the equation. b.) Find the total cost for a sale of 6 CDs. Slope-Intercept Form y = mx + b – where m is the slope – b is the y-intercept Your Turn- Identify the slope & y-intercept 1. y = 3x − 5 2. y = ½ x + 3 3. y = -2x + ¾ Slope-Intercept Form y = mx + b – where m is the slope – b is the y-intercept Your Turn- Identify the slope & y-intercept 1. y = 3x − 5 The slope is 3; the y-intercept is −5 2. y = ½ x + 3 The slope is ½ ; the y-intercept is 3 3. y = -2x + ¾ The slope is -2; the y-intercept is ¾ Using Slope & One Point When given one point and the slope use the point in the place of x & y to find the equation. m = ¼; ( 2, 7) 7 = ¼ (2) + b Using Slope & One Point Your Turn – 1. 2. 3. 4. Using Slope & One Point Your Turn – 1. 2. 3. 4. Using Two Points to Write an Equation • When given two points first find the slope • (x1, y1) (x2, y2) • (2, 4) ( -1 , 3) • Then plug in the slope and either point to find equation of the line. • 4 = ⅓(2) + b • 4= ⅔+b • 4-⅔=⅔-⅔+b • 3 ⅓ = b or ( ) Point-Slope Form of a Linear Equation The point-slope form of the equation of a non-vertical line that passes through the point (x₁, y₁) with slope m is: y - y₁ = m (x - x₁). • Suppose you know that a line passes through the point (3, 4) with slope 2. –You can quickly write an equation of the line using the x- and y-coordinates of the point and the slope. Graphing Using Point-Slope Form Graph the equation : y−5= (x − 2) Writing an Equation in Point-Slope Form Use the slope and one point on the Line (x₁, y₁), m y - y₁ = m (x - x₁) m= ½, ( 2, 3) y – 3 = ½ ( x – 2) Your Turn: 1. m= ⅔, ( -1, 4) 2. m= ⅓, ( -2, -5) 3. m= -2, ( 5, 3) Writing an Equation in Point-Slope Form Use the slope and one point on the Line (x₁, y₁), m y - y₁ = m (x - x₁) m= ½, ( 2, 3) y – 3 = ½ ( x – 2) Your Turn: 1. m= ⅔, ( -1, 4) y – 4 = ⅔ ( x – (-1)) 2. m= ⅓, ( -2, -5) y – (-5) = ⅓ ( x – (-2)) 3. m= -2, ( 5, 3) y – 3 = -2 (x – 5) Standard Form of a Linear Equation The standard form of a linear equation is Ax + By = C – where A, B, and C are real numbers – and A and B are not both zero. If the equation contains fractions or decimals, multiply to write the equation using integers (positive or negative whole numbers). •To get rid of fractions, multiply through with the denominator. •To get rid of decimals, multiply through by 10’s. (10, 100, 1000, etc) Standard Form of a Linear Equation Your Turn – 1. 2. 3. Standard Form of a Linear Equation Your Turn – 1. 2x – 3y = -15 2. 3x + 4y = -16 3. 4x + 10 y = 1 Special Graphs Absolute Value Equations are shaped like a V Graph the function y = | x | + 1 Quadratic Equations are shaped like a U Graph the function f(x) = x2 + 1. Translations A shift of a graph horizontally, vertically, or both. The result is a graph of the same shape and size, but in a different position. Below are the graphs of y = | x | and y = | x | + 2. Describe how the graphs are the same and how they are different. The graphs are the same shape. The y-intercept of the first graph is 0. The y-intercept of the second graph is 2. Graphing Translations Graphing a Horizontal Translation Graph each equation by translating y = | x |. Graphing Vertical Translations Graph y = | x | − 1. Start with the graph of y = | x |. Translate the graph down 1 unit. Graphing Translations Graphing a Horizontal Translation Graph each equation by translating y = | x |. Graphing Vertical Translations Graph y = | x | − 1. Start with the graph of y = | x |. Translate the graph down 1 unit. Writing an Absolute Value Equation A V-shaped graph that points upward or downward is the graph of an absolute value equation. Write the equation for a vertical translation Write the equation for a horizontal translation Writing an Equation for a Trend Line Make a scatter plot of the data at the left. Draw a trend line and write its equation. Use the equation to predict the wingspan of a hawk that is 28 in. long. Lines of Best Fit Using a Graphing Calculator The trend line that shows the relationship between two sets of data most accurately is called the line of best fit. A graphing calculator computes the equation of a line of best fit using a method called linear regression. The graphing calculator also gives you the correlation coefficient r, which tells you how closely the equation models the data. When the data points cluster around a line, there is a strong correlation between the line and the data. So the nearer r is to 1 or −1, the more closely the data cluster around the line of best fit.