MINI LESSON. Lesson 1b Linear Equations

Transcription

1 MINI LESSON Lesson 1b Linear Equations Lesson Objectives: 1. Identify LINEAR EQUATIONS 2. Determine slope and y-intercept for a LINEAR EQUATION 3. Determine the x-intercept for a LINEAR EQUATION 4. Draw graphs of LINEAR EQUATIONS 5. Graph Horizontal and Vertical lines 6. Write LINEAR EQUATIONS The topic of linear equations should be at least slightly familiar to students starting Intermediate Algebra. The basics are covered here with reminders of important ideas and concepts that will be heavily utilized in the next lesson. What is a Linear Equation? A LINEAR EQUATION is an equation that can be written in the form: y = mx + b with slope, m, and y-intercept (0, b) This is the SLOPE-INTERCEPT form for the equation of a line. Slope The slope of a line is denoted by the letter m. Given any two points, (x1, y1), (x2, y2), on a line, the slope is determined by computing the following ratio: m = y 2! y 1 x 2! x 1 Note: Slope is also referred to as the change in y over the change in x and is a measure of steepness and direction for a given line. Problem 1 WORKED EXAMPLE DETERMINE SLOPE OF A LINEAR EQUATION Find the slope of the line through the points (2, -5) and (-3, 4). m = 4! (!5)!3! (2) = 4 + 5!5 = 9!5 =! 9 5 Note: The slope is negative indicating the line decreases from left to right. If slope is positive, then the line increases from left to right. 1

2 Y-intercept Also called the VERTICAL INTERCEPT, this is the special ordered pair with coordinates (0, b). 0 is the value of x (input) and the resulting output (b) is the y-coordinate of the y-intercept. The y-intercept is often used to help when graphing a linear equation and/or to determine the initial output value in an application setting. Problem 2 WORKED EXAMPLE DETERMINE Y-INTERCEPT FOR A LINEAR EQUATION Find the y-intercept (also called the vertical intercept) for the equation y = -2x + 6. Method 1: Read the value of b from y=mx + b form. This equation is written in the form y = mx + b. Therefore, the y-intercept is (0, 6). If the equation is NOT in y = mx + b form, isolate y in the equation to make it so. Method 2: Solve for y when x = 0 Replace x with 0 to get y = -2(0) + 6 = = 6. The ordered pair (0,6) is the y-intercept. Problem 3 MEDIA EXAMPLE DETERMINE SLOPE/Y-INTERCEPT FOR LINEAR EQUATN Complete the table below. Note: Method 1 will be used to find the y-intercept for all the problems below because we are also asked to determine the slope. Practice finding the y-intercept using Method 2 as well. a) y = -2x +5 Equation y = mx + b form Slope Y-intercept b) y = 2 x c) y = 3 4 x + 2 d) 2x y = 4 e) 3x + 2y = 6 f) 8x = 4y 2

3 X-intercept Also called the HORIZONTAL INTERCEPT, this is the special ordered pair with coordinates (a, 0). 0 is the value of y (output) and the resulting input (a) is the x-coordinate of the x-intercept. The x-intercept is often used to help when graphing a linear equation and/or to determine the final input value in an application. Problem 4 WORKED EXAMPLE FIND X-INTERCEPT FOR A LINEAR EQUATION Find the x-intercept (also called the horizontal intercept) for the equation y = -2x + 6. Method: Replace the value of y with 0 then solve for the value of x. 0 = -2x = -2x -6/-2 = x 3 = x The x-intercept is (3, 0). Problem 5 MEDIA EXAMPLE FIND X-INTERCEPT FOR A LINEAR EQUATION For each of the following problems, determine the x-intercept as an ordered pair using the methods above. Equation Show Work a) y = -2x +5 b) y = 2 x c) y = 3 4 x + 2 d) 2x y = 4 e) 3x + 2y = 6 f) 8x = 4y 3

4 Problem 6 YOU TRY DRAW GRAPHS OF LINEAR EQUATIONS Use the equation y =! 3 x + 9 for all parts of this problem. Label all plotted points. 2 a) Use the x-intercept and y-intercept to draw the graph of the line. Show your work to find these points. PLOT and LABEL the intercepts on the graph then connect them to draw your line. x-intercept : (, ) y-intercept: (, ) Work to find x-intercept Work to find y-intercept b) Determine the coordinates of two OTHER ordered pairs and use those to graph the line. PLOT and LABEL the points you use. Remember from your previous math work that you can create a t-table with ordered pairs other than the intercepts OR you can insert the equation into your Y= list on your TI 83/84 calculator then use the TABLE feature to identify two other ordered pairs. Remember to enter as y1=(-3/2)x + 9. Ordered pair 1: (, ) Ordered pair 2: (, ) NOTICE that your graphs for parts a) and b) should look exactly the same. 4

5 Special Linear Equations The following media problem will introduce you to two special types of linear equations: horizontal and vertical lines. Problem 7 MEDIA EXAMPLE GRAPHING HORIZONTAL/VERTICAL LINES a) Use the grid below to graph the equation y = -2. Identify the slope and y-intercept. b) Use the grid below to graph the equations x = 5. Identify the slope and y-intercept. Equations of Vertical Lines equation: x = a x-intercept: (a, 0) y-intercept: none slope: m is undefined Equations of Horizontal Lines equation: y = b x-intercept: none y-intercept: (0, b) slope: m = 0 5

6 Writing Equations of Lines Critical to a thorough understanding of linear equations is the ability to write the equation of a line given different pieces of information. The following process will work for almost every situation you are presented with and will be illustrated several times in the media problems to follow. Step 1: Determine the value of the slope, m. Step 2: Determine the coordinates of one ordered pair. Step 3: Plug the values for the ordered pair, and the value for the slope, into y = mx + b Step 4: Solve for b Step 5: Use the values for m and b to write the resulting equation in y = mx + b form. Problem 8 MEDIA EXAMPLE WRITING EQUATIONS OF LINES For each of the following, find the equation of the line that meets the following criteria: a) Slope m = -4 passing through the point (0, 3). b) Passing through the points (0, -2) and (1, 5) c) Passing through the points (-2, -3) and (4, -9) d) Parallel to y = 3x 7 and passing through (2, -5) e) Passing through (2, 4) with an x-intercept of -2. f) Vertical line passing through (-3, 5). g) Horizontal line passing through (-3, 5). 6

7 Problem 9 YOU TRY WRITING LINEAR EQUATIONS FROM GRAPHS Use the given graph to help answer the questions below. Assume the line intersects grid corners at integer (not decimal) values. a) Is the line above increasing, decreasing, or constant? Read the graph from left to right. b) What is the vertical (y) intercept? y-intercept? Also, plot, label on the graph. c) What is the horizontal (x) intercept? d) What is the slope (m)? Show work at right to compute. Hint: Use two points from the graph. x-intercept? Also, plot, label on the graph. Work to find slope: Slope = e) What is the equation of the line in y=mx + b form? Hint: Use the slope and y-intercept from above. Equation of the line: 7

8 Problem 10 YOU TRY WRITING EQUATIONS OF LINES a) Find the equation of the line passing through the points (1,4) and (3,-2) and write your equation in the form y = mx + b. Show complete work in this space. b) What is the vertical (y) intercept for this equation? Show work or explain your result. c) What is the horizontal (x) intercept for this equation? Show complete work to find this. Problem 11 YOU TRY HORIZONAL AND VERTICAL LINES a) Given the ordered pair (2, -3) Write the equation of the vertical line through this point. Identify the slope of the line: What is the y-intercept? What is the x-intercept? b) Given the ordered pair (2, -3) Write the equation of the horizontal line through this point. Identify the slope of the line: What is the y-intercept? What is the x-intercept? 8

9 Problem 12 WORKED EXAMPLE WRITING EQUATIONS OF LINES Write an equation of the line to satisfy each set of conditions. a) Line contains the points (-3, 5) and (0, 1) Slope y-intercept Equation Use the ordered pairs (-3, 5) and (0, 1) to compute slope. Plug m and b into y = mx + b m = 1" 5 0 " ("3) = "4 3 Given y-intercept in the ordered pair (0, 1) that was provided then b = 1. y = " 4 3 x + 1 b) Line contains points (-4, -3) and (2, 6) Slope y-intercept Equation Use the ordered pairs (-4, -3) and (2, 6) to compute slope. Plug m and b into y = mx + b m = 6! (!3) 2! (!4) = = 9 6 = 3 2 c) Line has the following graph: Not given y-intercept. Pick one given ordered pair and plug m and ordered pair into y = mx+b. Solve for b. Using (2, 6) then 6 = 3 2 (2) + b so 6 = 3 + b or b = 3. y = 3 2 x + 3 Slope y-intercept Equation Identify two ordered pairs from the graph and use them to Read the y-intercept from the graph. Ordered pair is (0, -3). Plug m and b into y = mx + b determine the slope. Therefore b = -3. y = 1 x (5, 0) and (3, -1) m =!1! (0) 3! (5) =!1!2 = 1 2 9

10 ANSWERS YOU TRY PROBLEMS You Try Problem 6: x-intercept (6, 0), y-intercept (0, 9), Answers vary on other ordered pairs, graphs for parts a) b) should be the same You Try Problem 9: a) decreasing b) (0, 4) c) (4, 0) d) slope = -1 e) y = -x + 4 You Try Problem 10: a) y = -3x + 7 b) (0,7) c) (7/3, 0) or (2.33, 0) You Try Problem 11: a) equation x = 2, slope is undefined, no y-intercept, x-intercept (2, 0) b) equation y = -3, slope m = 0, y-intercept (0, -3), no x-intercept 10