Name Period Date LF5.1 Slope-Intercept Form Graph lines. Interpret the slope of the graph of a line. Find equations of lines. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane. LF5.2 Equations of Lines in Different Forms Understand the standard form of a linear equation. Understand the point-slope form of a line Change one form of a linear equation to other forms LF5.3 Graphing Inequalities in Two Variables Understand that the boundary line of a linear inequality is represented by a linear equation Understand that the graph of a linear inequality is a half-plane. LF5 STUDENT PAGES LINEAR FUNCTIONS STUDENT PACKET 5: INTRODUCTION TO LINEAR FUNCTIONS LF5.4 Vocabulary, Skill Builders, and Review 21 1 7 17 LF5 SP
WORD BANK Word or Phrase Definition or Explanation Example or Picture boundary line explicit rule half plane linear function linear inequality point-slope form of a linear equation slope of a line slope-intercept form of a linear equation standard form of a linear equation x-intercept y-intercept LF5 SP0
5.1 Slope-Intercept Form Ready (Summary) We will find equations of lines in slopeintercept form. We will extend the meaning of slope to horizontal and vertical lines. We will use properties of parallels and similar triangles to deepen our understanding of the meaning of slope of a line. Label some points on this line. SLOPE-INTERCEPT FORM Go (Warmup) Set (Goals) Graph lines. Interpret the slope of the graph of a line. Find equations of lines. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane. 1. When x = 0, then y =. This is called the. 2. Select two points on the line. Find the from one point to another. difference in the y-coordinates difference in the x-coordinates This is called the. y as you move x LF5 SP1
5.1 Slope-Intercept Form B FINDING EQUATIONS OF LINES 1. Write the coordinates next to the labeled points. K A I F Slope-intercept form of a line: 1. For Line AF Slope: y-intercept: M Equation: y D N C E H x G LF5 SP2
5.1 Slope-Intercept Form FINDING EQUATIONS OF LINES (continued) Find the slope, the y-intercept, and the equation in slope-intercept form for each line on the previous page. 1. Line BC Slope: y-intercept: Equation: 3. Line IK Slope: y-intercept: Equation: 2. Line DE Slope: y-intercept: Equation: 4. Line HG Slope: y-intercept: Equation: LF5 SP3
5.1 Slope-Intercept Form FINDING MORE EQUATIONS Graph a line that fits each of these descriptions. Find the slope, the y-intercept, and the equation of each line in slope-intercept form. Use your equation to determine if a particular point lies on the line. Find the x-intercept (the point where the graph crosses the x-axis, or x-value when y = 0) y 1. Graph the line that goes through the origin and the point (5, 6). Slope: y-intercept: Equation: Use your equation to show that the point (-5,-6) lies on the line. x-intercept: 2. Graph the line that goes through (-1, 2) and has a slope of 2 Slope: y-intercept: Equation: Use your equation to show that the point (1, 2) does not lie on the line. x-intercept: y x x LF5 SP4
5.1 Slope-Intercept Form FINDING MORE EQUATIONS (continued) Graph a line that fits each of these descriptions. Find the slope, the y-intercept, and the equation of each line in slope-intercept form. Use your equation to determine if a particular point lies on the line. Find the x-intercept. y 3. Graph the line that goes through the points (2, 1) and (-2, 3). Slope: y-intercept: Equation: Use your equation to show that the point (2, 4) does not lie on the line. x-intercept: 4. Graph the line that has intercepts (0, -1) and (-4, 0). Slope: y-intercept: Equation: Use your equation to show that the point (4,-2) lies on the line. x-intercept: y x x LF5 SP5
5.1 Slope-Intercept Form 1. 2. 3. 4. Line PQ WY LM RS HORIZONTAL AND VERTICAL LINES R Two points on the line (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) W Y P S y x-intercept y-intercept slope equation 5. What is the slope of a horizontal line?. Can you write the equation of a horizontal line in slope-intercept form? Explain. 6. What is the slope of a vertical line?. Can you write the equation of a vertical line in slope-intercept form? Explain. Q L M x y = x = LF5 SP6
5.2 Equations of Lines in Different Forms EQUATIONS OF LINES IN DIFFERENT FORMS Ready (Summary) We will review the slope-intercept form of a linear function and learn about two other forms of this equation: standard form, and point-slope form. 1. Here is an equation in slope-intercept form. 2 y =- x +1 3 a. Slope: m = b. The y-intercept is c. Graph the equation below Go (Warmup) Set (Goals) Understand the standard form of a linear equation Understand the point-slope form of a linear equation Change one form of a linear equation to other forms 2. Here is a graph of a linear equation. a. Slope: m = b. The y-intercept is c. The equation in slope-intercept form is y = LF5 SP7
5.2 Equations of Lines in Different Forms STANDARD FORM The standard form of a linear equation is ax + by = c, where a and b cannot both be equal to zero. Write the standard form and slope-intercept form for each equation below. Note that it may already be in one of these forms. 1. y = 3x + 2 slope-intercept form: standard form: 3. x = 2 4y slope-intercept form: standard form: 2. 2x + y = 6 slope-intercept form: standard form: 4. 3x + 1 2 y 1 = 0 slope-intercept form: standard form: LF5 SP8
5.2 Equations of Lines in Different Forms REVISITING SLOPE: NEW NOTATION Recall that the slope of a line (represented by m in the equation y = mx + b) is the ratio of the vertical change to the horizontal change, often referred to as rise over run. m verticalchange changein y Δy ( y 2 - y 1) = = = = horizontalchange changein x Δ x ( x - x ) 2 1 is the Greek letter delta, and here it stands for the words change in. In figures 1 and 2 note the highlighted points and the directions of the dashed arrows. 1. Figure 1 2. Figure 2 a. count the vertical change (up) (down) b. count the horizontal change (right) (left) c. write the slope m Figure 1 (5, 4) verticalchange = = horizontalchange Δy Δx m = m = 3. Are the slope ratios you found in problems 1 and 2 equivalent or not? Explain. 4. What does (1, -2) Δy Δx mean? Figure 2 (5, 4) (1, -2) LF5 SP9
5.2 Equations of Lines in Different Forms REVISITING SLOPE: NEW NOTATION (continued) A common way to refer to unknown coordinates is to use subscript notation. Refer to one point as (x 1, y 1 ). Refer to another point as (x 2, y 2 ). The subscripts (the small 1 and 2) are only for naming purposes. (x 2, y 2 ) y (x 1, y 1 ) Find the slope of the line given the points from the previous page, (5, 4) and (1, -2). Note: either point can be named (x 1, y 1 ) or (x 2, y 2 ). a. difference between y-coordinates ( y = y 2 y 1 ) b. difference between x-coordinates ( x = x 2 x 1 ) c. Δy m = = y y Δ x x x 2 1 2 1 5. (x 2, y 2 ) = (5, 4) (x 1, y 1 ) = (1, -2) 6. (x 2, y 2 ) = (1, -2) (x 1, y 1 ) = (5, 4) 4 ( ) = ( ) ( ) = 5 ( ) = ( ) ( ) = m = m = 7. Are the ratios you found in problems 5 and 6 equivalent or not? Explain. 8. The points (3, 2) and (0, 6) lie on a line. What is incorrect about the following slope calculation? Δy 6-2 4 m = = = Δx 3-0 3 9. Explain why lining up the points in the following fashion might help to avoid the mistake made above. (x 2, y 2 ) = (0, 6) (x 1, y 1 ) = (3, 2) x LF5 SP10
5.2 Equations of Lines in Different Forms SLOPE: NEW NOTATION PRACTICE Given the following points on a line, (-1, 3) and (5, -5), find: a. difference between y-coordinates ( y = y 2 y 1 ) b. difference between x-coordinates ( x = x 2 x 1 ) c. Δy m = = y y Δ x x x 2 1 2 1 1. (x 2, y 2 ) = (-1, 3) (x 1, y 1 ) = (5, -5) ( ) ( ) = ( ) ( ) = For the given pairs of points on a line, find the slope. m = m = 2. (x 2, y 2 ) = (5, -5) (x 1, y 1 ) = (-1, 3) ( ) ( ) = ( ) ( ) = (x 2, y 2 ) 3. (2, 5) 4. (-1, 6) 5. (0, 3) (x 1, y 1 ) (-2, 3) (0, -4) (-2, 5) Δy m = Δ x = y y x x 2 1 2 1 ( ) m = = ( ) 6. Two points on a line are (-5, 10) and (3, -6). ( ) m = = ( ) Δy ( y 2 - y 1) Place one ordered pair over the other to find m = = Δ x ( x - x ) 7. Jerome was given two points on a line, (-3, 0) and (4, -1). To calculate the slope he did the following: 2 1 ( ) m = = ( ) 4 -(-3) 7 =. What was his mistake? -1-(0) -1 LF5 SP11
5.2 Equations of Lines in Different Forms The slope of the line to the right is 2 One point identified on the line is (1, 4) It is not obvious what the other highlighted point is, so we will call it (x, y) 1. 2. Fill in the blanks below to verify: the point-slope form of a linear equation: y y 1 = m(x x 1 ) Δy y 2 - ( ) m = = Δ x x - ( ) 2 POINT-SLOPE FORM the slope formula ( y )-( ) 2= substitution: m = 2; ( )-( ) 3. (2)( ) = ( ) (x 2, y 2 ) = (x, y) (x 1, y 1 ) = (1, 4) multiplication property of equality: multiply both sides by (x 1) 4. 2( x -1) = ( ) simplify right side by multiplication 5. = 2( x - 1) 6. Look at the equation in problem. a. Find the slope (m) and draw a small arrow pointing to it. symmetric property of equality: expressions switch sides (This equation is now in point-slope form.) b. Find the known point on the line, (x 1, y 1 ) = (1, 4), and underline each coordinate value. c. Find the unknown point on the line, (x 2, y 2 ) = (x, y), and circle each coordinate value. d. Write the equation of this line in slope-intercept form. (1, 4) (x, y) LF5 SP12
5.2 Equations of Lines in Different Forms POINT-SLOPE FORM (continued) 7. The point-slope form of a linear equation is 8. The standard form of a linear equation is 9. The slope-intercept form of a linear equation is Write the equations of the lines in the following forms in any order desired. 10. Given: m = -3 and one point on the line is (5, 4) 11.Given: m = 5 and one point on the line is (- 12.m = 1 and one point on the line is (0, -5) 2, -6) point-slope form: point-slope form: point-slope form: slope-intercept form: slope-intercept form: slope-intercept form: standard form: standard form: standard form: LF5 SP13
5.2 Equations of Lines in Different Forms PRACTICE WITH DIFFERENT FORMS Find equations of lines in different forms. Use the information given. 1. Given: 2. Given: m = -2; y-intercept is 3 slope-intercept form point-slope form standard form (-2, 3) is on the line; slope = 1 2 slope-intercept form point-slope form standard form LF5 SP14
5.2 Equations of Lines in Different Forms PRACTICE WITH DIFFERENT FORMS (continued Find equations of lines in different forms. Use the information given. 3. Given: 4. Given (table): (4, 3) and (-4, -5) are on the line slope-intercept form point-slope form standard form x y 0-6 1-3 2 0 3 3 4 6 slope-intercept form point-slope form standard form LF5 SP15
5.2 Equations of Lines in Different Forms PRACTICE WITH DIFFERENT FORMS (continued) Find equations of lines in different forms. Use the information given. 5. Given: 6. Given: (0, -2) and (-2, 0) are on the line slope-intercept form point-slope form standard form slope-intercept form point-slope form standard form LF5 SP16
5.3 Graphing Inequalities in Two Variables GRAPHING INEQUALITIES IN TWO VARIABLES Ready (Summary) We will graph linear inequalities in two variables by graphing dashed or solid boundary lines and then shading halfplanes appropriately. Go (Warmup) Set (Goals) Understand that the boundary line of a linear inequality is represented by a linear equation Understand that the graph of a linear inequality is a half-plane Fill in the table below by writing the inequality in symbols, graphing it, and testing a number. These inequalities are in one variable. 1. -4 is greater than x 2. Words Symbols Graph Test a Number The opposite of x is less than or equal to -2 3. On a graph, what does a closed dot ( ) mean? 4. On a graph, what does an open dot ( ) mean? 5. When solving an inequality, doing what to both sides causes the inequality symbol to change direction? LF5 SP17
5.3 Graphing Inequalities in Two Variables INEQUALITIES IN TWO VARIABLES: INTRODUCTION 1. Graph the line y = x to the right. 2. Write some points below in which the y-coordinate is greater than the x-coordinate. (Use simple numbers that are between -5 and 5.) 3. Graph a few of these points. Describe where they lie in relation to the y = x line. 4. Write some points below in which the y-coordinate is less than the x-coordinate. (Use simple numbers that are between -5 and 5.) 5. Graph a few of these points. Describe where they lie in relation to the y = x line. 6. Graph the inequality y x by first graphing y = x, and then lightly shading the region in which the y-coordinates are greater than the x-coordinates. The shaded region is called a half plane. Test an ordered pair by substituting the x and y values into the inequality. 7. Graph the inequality y x by first graphing y = x, and then lightly shading the region in which the y-coordinates are less than the x-coordinates. The shaded region is called a half plane. Test an ordered pair by substituting the x and y values into the inequality. LF5 SP18
5.3 Graphing Inequalities in Two Variables INEQUALITIES IN TWO VARIABLES: PRACTICE 1 When graphing an inequality, its boundary line is solid when all of the points on the line are included in the solution set of the inequality (similar to the closed dot on a number line). The boundary line is dashed when the points on the line are not included (similar to the open dot on a number line). Graph each inequality (slope-intercept form of a line tends to be easier to graph). Consider the equation of the boundary line and graph it as a solid or dashed line. Shade the appropriate half-plane. Test at least one point by substituting it into the inequality. 1. 2x y > 4 3. y < 2(x 1) 2. y - 1 3 x 1 4. y 3 1 (x 4) 2 LF5 SP19
5.3 Graphing Inequalities in Two Variables INEQUALITIES IN TWO VARIABLES: PRACTICE 2 Write an inequality to match each graph. 1. 2. 3. 4. 5. 6. LF5 SP20
5.4 Vocabulary, Skill Builders, and Review Across 5 line that separates plane into two half planes FOCUS ON VOCABULARY Down 1 a number that describes the slant of a line 7 another name for input-output rule 2 a form of a linear equation: y 2 y 1 = m(x 2 x 1 ) (2 words) 10 a form of the linear function of the form: y = mx + b (2 words) 11 a form of linear equation: ax + by = c 3 2x 3y > 5 is an example of a linear 4 another name for the shaded portion of a linear inequality 6 (0, 7) is an example of a(n) - intercept 8 (-8, 0) is an example of a(n) - intercept 9 a function whose graph is a line LF5 SP21
5.4 Vocabulary, Skill Builders, and Review Complete. Draw line segments with the following slopes. 1. Line AB with a slope of 2 1. 2. Line CD with a slope of -1 3. 3. Line LM with a slope of 1 4. 4. Line PQ with a slope of -3. Given each set of ordered pairs, find the slope of the line that goes through them. 1. E(2, -1) and F(7, 3) 2. G(-1, 1) and H(-2, 8) 3. J(-1, -3) and K(-7, 1) 4. N(2, 6) and R(7, 6) SKILL BUILDER 1 A C L P LF5 SP22
5.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 2 1. The lengths of the sides of a pentagon are consecutive odd numbers. The perimeter is 435 cm. Find the length of each side of the pentagon. 2. Arnon and Bob are taking the train from Los Angeles to San Francisco. Arnon leaves the train station in Los Angeles at 8:00 AM on a slow train traveling 40 mph. Bob leaves Los Angeles at 1 PM on a fast train traveling 75 mph. If it is 400 miles from Los Angeles to San Francisco, who arrives first? Show your work and explain your reasoning. LF5 SP23
5.4 Vocabulary, Skill Builders, and Review 1. Draw a line through point A (1, 2) with a slope of 2 1. What is the y-intercept? 2. Draw a line through point B (-2, -2) with a slope of -1 3. What is the x-intercept? 3. Draw a line through point C (4, 4) with a slope of 1 4. Name a point on this line that is in the 2 nd quadrant. SKILL BUILDER 3 Given each set of ordered pairs, use the slope formula to find the slope of the line that goes through them. 5. S(5, 4) and T(2, 3) 6. U(2, 10) and V(5, 1) 7. W(10, 16) and X(-2, -4) 8. Y(-5, -8) and Z(0, -12) LF5 SP24
5.4 Vocabulary, Skill Builders, and Review Compute. SKILL BUILDER 4 1. Slope of AB: 2. Slope of AC: 3. Slope of BC: 4. Slope of DB: 5. Label and identify three similar right triangles using a portion of the line as the hypotenuse and horizontal and vertical segments for legs. What is true about the ratio of corresponding legs in these triangles? 6. What do you notice about the slope of each line segment? 7. Locate a point F so that the slope of line EF = -2. Then draw line EF. 1 8. Name these points on your line. E (, ) F (, ) G ( 0, ) H (, 0 ) 9. Find the slope of line EH. 10. Find the equation of line EH in slope-intercept form. y = Use for problems 1-6: A B C Use for problems 7-10: E y D x LF5 SP25
5.4 Vocabulary, Skill Builders, and Review 1. Two points on the line Slope y-intercept Equation of the line in the form y = mx + b 2. Two points on the line Slope y-intercept Equation of the line in the form y = mx + b (4, -2) and (-4, 0) (-2, -3) and (1, -6) SKILL BUILDER 5 3. Complete the table, graph the values, and write an equation for the line that fits the data in slope-intercept form. There are 4 quarts in 1 gallon. Gallons (x) 0 1 1.5 2 2.75 3 x Quarts (y) 0 4 6 y = 4. Does connecting your points with a line make sense? Explain. LF5 SP26
5.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 6 Draw the following lines on the coordinate axes above. Then fill in the table. two points equation of the line in slope y-intercept x-intercept on the line slope-intercept form 1. (-3, 1) (1, 5) 2. (-3, 9) 0 3. (-2, 4) 1 4. 5. (-3, -1) (6, -4) 1-2 2 LF5 SP27
5.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 7 Draw the following lines on the coordinate axes above. Then fill in the table. One point equation of the line in slope y-intercept x-intercept on the line slope-intercept form 1. (1, 1) -3 2. (4, -3) 3 3. 2 1-2 4. y = -3x + 4 5. y = 1-3 x 4 LF5 SP28
5.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 8 Find equations of lines in different forms. Use the information given. Given 1. y- intercept is -2 slope = 4 3 2. slope = -3; (1, -1) is on the line; 3. (3, -1) and (6, 1) are on the line Form of linear equation slope-intercept standard point-slope LF5 SP29
5.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 9 Find equations of lines in different forms. Use the information given. Given 1. table x y 0-1 1 1 2 3 3 5 4 7 2. Graph 3. (2, -2) and (-2, 2) are on the line Form of linear equation Slope-intercept Standard Point-slope LF5 SP30
5.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 10 Write each statement using symbols. If the variable is on the right side, change it to the left side using appropriate properties. Then graph each. 1. x is equal to -1 2. x is greater than -3 3. the opposite of x is less than or equal to -2 4. 4 is greater than x 5. -1 is less than or equal to the opposite of x. Graph each inequality. Be sure they are in slope-intercept form first. 2 7. -5 y+ x 10 6. y > x -1 3 8. Describe the differences between the graph of an inequality in one variable and the graph of an inequality in two variables. LF5 SP31
5.4 Vocabulary, Skill Builders, and Review TEST PREPARATION Show your work on a separate sheet of paper and choose the best answer. 1. Find the slope of the line through the points (0, 3) and (-5, 0). A. 3 5 B. 3 - C. 5 2. Which of the following best describes the slope of the line through the points (-3, 2) and (-3, -3)? A. Positive slope B. Negative slope C. Zero slope D. No slope 3. Which of these equations represents the line through the points (-5, 13) and (5, 3)? A. y = -x 8 B. y = x + 8 C. y = x 8 D. y = -x + 8 4. Which equation is equivalent to y-5 = 3( x - 1) and is also in standard form? A. y = 3x 2 B. y = 3x + 2 C. -3x + y = -2 D. 3x y = -2 5. Which statement about linear inequalities in NOT true? A. The graph of a linear inequality is a half-plane. B. The boundary line of a linear inequality must be either dashed or solid. C. If the boundary line of a linear inequality is dashed, the points on the boundary line are solutions to the inequality. D. If a point on one side of the boundary line of a linear inequality is a solution, a point on the other side is NOT a solution. 5 3 D. 5-3 LF5 SP32
5.4 Vocabulary, Skill Builders, and Review KNOWLEDGE CHECK Show your work on a separate sheet of paper and write your answers on this page. 5.1: Slope Intercept Form Find the equation of each line in slope-intercept form. 1. A line through the point (-1, -1) with a slope of 3. 2. A line with an x-intercept of -2 and a y-intercept of -4. 5.3: Equations of Lines in Different Forms 3. Write the equations from problems 1 and 2 above in: a. Point-slope form b. Standard form 4. For the given input-output table to the right: a. Write the equation in slope-intercept form b. Write the equation in standard form c. Write the equation in point-slope form d. Graph the equation 5.4: Graphing Inequalities in Two Variables 5. Graph each inequality. a. y - x +3 b. x- y < -4 x y 0-2 1 1-1 2 2-1 3 1-2 4 0 LF5 SP33
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HOME-SCHOOL CONNECTION Here are some questions to review with your young mathematician. Use graph paper as needed for problems 1-4 to find the equation of each line in (a) slopeintercept form, (b) point-slope form, and (c) standard form. 1. The line through the point (0, -2) with a slope of 4. 2. The line through the points (-3, 3) and (-2, 1). 3. The line with a slope of 1 - and an x-intercept of 5. 5 4. The line through the points (3, 1) and (-5, 1). 5. Graph 4 x+2y -6 Parent (or Guardian) signature LF5 SP37
COMMON CORE STATE STANDARDS MATHEMATICS 8.EE.6 8.F.2 A-CED-4 A-REI-10 A-REI-3 A-REI-12 F-IF-7a SELECTED COMMON CORE STATE STANDARDS FOR MATHEMATICS Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V = IR to highlight resistance R. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Graph the solutions to a linear inequality in two variables as a half plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes Graph linear and quadratic functions and show intercepts, maxima, and minima. STANDARDS FOR MATHEMATICAL PRACTICE MP1 Make sense of problems and persevere in solving them. MP2 Reason abstractly and quantitatively. MP3 Construct viable arguments and critique the reasoning of others. MP4 Model with mathematics. MP5 Use appropriate tools strategically. MP6 Attend to precision. MP7 Look for and make use of structure. MP8 Look for and express regularity in repeated reasoning. LF5 SP38