1.7 Solve Linear-Quadratic Systems Marina is a set designer. She plans movie sets using freehand sketches and her computer. In one scene, a banner will hang across a parabolic archway. To make it look interesting, she has decided to put the banner on an angle. She sets the banner along a line defined by the linear equation y 5 0.24x 7.2, with x representing the horizontal distance and y the vertical distance, in metres, from one foot of the archway. The archway is modelled by the quadratic equation y 5 0.48x2 4.8x. How can Marina use the equations to determine the points where the banner needs to be attached to the archway and the length of the banner? In this section, you will develop the tools needed to help Marina with these calculations. Tools Investigate A • grid paper Optional How can a line and a parabola intersect? • graphing calculator Work with a partner. 1. Consider a line and a parabola. At how many points could they intersect? Draw sketches to illustrate your answer. 2. Create pairs of equations for each possibility that you identified in step 1. Use algebraic reasoning to show that your examples are correct. 3. In your algebraic reasoning in step 2, you will have solved a quadratic equation for each situation. Compute the value of the discriminant for each example. 4. Reflect Describe how you can predict the number of points of intersection of a linear function and a quadratic function using algebraic reasoning. 60 MHR • Functions 11 • Chapter 1 Investigate B Tools • grid paper How can you connect the discriminant to the intersection of a linear or and a quadratic function? • graphing calculator In this Investigate, you will create the equations of lines with slope 2 that intersect the quadratic function y 5 x2 4x 4. 1. Write a linear function, in slope y-intercept form, with slope 2 and an unknown y-intercept represented by k. 2. Eliminate y by substituting the expression for y from the linear equation into the quadratic equation. Simplify so you have a quadratic equation of the form ax2 bx c 5 0. 3. Substitute the values or expressions for a, b, and c into the discriminant b2 4ac. 4. In Section 1.6, you learned that the discriminant determines the number of solutions for a quadratic equation. Take advantage of this fact to answer the following questions. a) What values of k will make the discriminant positive? How many points of intersection do the line and the quadratic have in this case? b) What values of k will make the discriminant zero? How many points of intersection do the line and the quadratic have in this case? c) What values of k will make the discriminant negative? How many points of intersection do the line and the quadratic have in this case? 5. Reflect With the solutions from step 4, write an equation for a linear equation, in slope y-intercept form y 5 mx b, with slope 2, that a) intersects the quadratic function at two points b) intersects the quadratic function at one point c) does not intersect the quadratic function 6. Verify each solution in step 5 by graphing the quadratic function y 5 x2 4x 4 and each of your linear functions. 1.7 Solve Linear-Quadratic Systems • MHR 61 Example 1 Find the Points of Intersection of a Linear-Quadratic System of Equations In the opening of this section, you were introduced to Marina. In a set design, she has a banner on an angle across an archway. She is working with the equations y 5 0.24x 7.2 and y 5 0.48x2 4.8x, where x represents the horizontal distance and y the vertical distance, both in metres, from one foot of the archway. a) Determine the coordinates of the points where the two functions intersect. b) Interpret the solutions in the context. Solution a) Method 1: Use Pencil and Paper Eliminate y by equating the two functions. 0.48x2 4.8x 5 0.24x 7.2 0.48x2 4.8x 0.24x 7.2 5 0 Rearrange the terms so the right side is zero. 0.48x2 4.56x 7.2 5 0 Simplify. 2x2 19x 30 5 0 Divide both sides by —0.24. 2x2 4x 15x 30 5 0 Use grouping to factor. 2x(x 2) 15(x 2) 5 0 (x 2)(2x 15) 5 0 Therefore, x 5 2 or x 5 7.5. Substitute into either function to find the corresponding values for y. The linear function is easier to use here. y 5 0.24x 7.2 For x 5 2: For x 5 7.5: y 5 0.24(2) 7.2 y 5 0.24(7.5) 7.2 5 7.68 5 9 The coordinates of the points where the two functions intersect are (2, 7.68) and (7.5, 9). 62 MHR • Functions 11 • Chapter 1 Method 2: Use a Graphing Calculator • Enter the two functions: Y1 5 0.48x2 4.8x and Y2 5 0.24x 7.2. • Use the window settings shown. • Press GRAPH . • Press 2nd [CALC]. • Use the Intersect operation to find the coordinates of each point of intersection. The coordinates of the points where the two functions intersect are (2, 7.68) and (7.5, 9). Method 3: Use a TI-NspireTM CAS Graphing Calculator Turn on the TI-NspireTM CAS graphing calculator. • Press c and select 6:New Document. • Select 2:Add Graphs & Geometry. • Type 0.48x2 4.8x for function f1. Press ·. • Type 0.24x 7.2 for function f2. Press ·. • Press b. Select 4:Window. • Select 6:Zoom – Quadrant 1. The graphs will be displayed. • Press b. Select 6:Points & Lines. • Select 3:Intersection Point(s). Move the cursor to the first graph and press ·. Move the cursor to the second graph and press ·. Press d. The coordinates of the points where the two functions intersect are displayed as (2, 7.68) and (7.5, 9). b) These solutions tell Marina that one end of the banner should be attached 2 m horizontally from the left foot of the arch and 7.68 m upward. The other end of the banner should be attached 7.5 m horizontally and 9 m upward. 1.7 Solve Linear-Quadratic Systems • MHR 63 Example 2 Determine Whether a Linear Function Intersects a Quadratic Function Determine algebraically whether the given linear and quadratic functions intersect. If they do intersect, determine the number of points of intersection. a) y 5 3x 5 and y 5 3x2 2x 4 b) y 5 x 2 and y 5 2x2 x 3 Solution a) Equate the expressions and simplify. 3x2 2x 4 5 3x 5 3x2 2x 4 3x 5 5 0 Rearrange the terms so the right side is zero. 3x2 5x 9 5 0 Simplify. a 5 3, b 5 5, and c 5 9. Use the discriminant: b2 4ac 5 (5)2 4(3)(9) 5 25 108 5 133 Since the discriminant is greater than zero, there are two solutions. This means that the linear-quadratic system has two points of intersection. b) Equate the expressions and simplify. 2x2 x 3 5 x 2 2x2 x 3 x 2 5 0 Rearrange the terms so the right side is zero. 2x2 2x 1 5 0 Simplify. a 5 2, b 5 2, and c 5 1. Use the discriminant: b2 4ac 5 22 4(2)(1) 5 4 8 5 4 Since the discriminant is less than zero, there are no solutions. This means that the linear-quadratic system has no points of intersection. 64 MHR • Functions 11 • Chapter 1 In this section, you have considered how a line can intersect a curve secant such as a quadratic function. One type of intersection results in a secant • a line that intersects and the other results in a tangent line to the quadratic function. a curve at two distinct points tangent line • a line that touches a Example 3 curve at one point and has the slope of the Determine the y-intercept for a Tangent Line to a Quadratic Function curve at that point If a line with slope 4 has one point of intersection with the quadratic secant function y 5 ​ _1 ​x2 2x 8, what is the y-intercept of the line? Write the 2 equation of the line in slope y-intercept form. Solution tangent The line can be modelled as y 5 4x k, where k represents the y-intercept. Substitute for y in the quadratic function: Connections _1 2 ​ ​ x 2x 8 5 4x k One of the main topics 2 that you will study in ​ _1 ​x2 2x 8 k 5 0 calculus is determining 2 the slope of a tangent to a curve at a point on ​ _1 ​x2 2x (8 k) 5 0 2 the curve. Then, a 5 ​ _1 ​, b 5 2, and c 5 8 k. 2 If the discriminant equals zero, there is only one root. Substitute into b2 4ac 5 0. (2)2 4 ​ _1 ​ (8 k) 5 0 ( 2 ) 4 2(8 k) 5 0 Technology Tip 4 16 2k 5 0 Sometimes a “friendly 2k 5 20 window” is needed to k 5 10 cause the calculator to display exact values. The y-intercept of the line that touches the quadratic at one point is 10. Choose multiples of The equation of the line is y 5 4x 10. 94 for the domain. To show the exact point This solution can be verified using a graphing calculator.