University:
High School
Course:
High School Functions
Academic year:

2024
Views:

182

Pages:

3
Author:

J. Anderson

[1.7] SOLVE LINEAR-QUADRACTIC SYSTEMSWHAT YOU NEED TO KNOWA linear function and a quadratic function may:Intersect at 2 points (the line is a secant)yIntersect at 1 point (the line is a tangent line)Never intersectThe discriminant can be used to determine which ofthe above situations occur.The quadratic formula can be used to determine theX-values of actual points ofintersection .MATH TERMS1. SECANTA line that intersect a Curve at two distinct points.secant2. TANGENT LINEA line that touches a curve at one point and has theslope of the curve at that point.tangentEXAMPLE #1: FIND THE POINTS OF INTERSECTION OF A LINEAR QUADRATIC SYSTEMf(x) = 0.24x + 7.2 & f(x) = -0.48x2 + 4.8xa. Determine the coordinates of the points where the two functions intersect.(5. 12)- - 0.48x2+4.8x = 0.24x +7,21060(7.5.- 0.48x 224.8x-0.24x - 7.2=0(2,7.68)5- 0.48y 2 +4.56v-7.2 =05(o, o)(10, 0 o)x= -((2)=0.24(2)+7.22(-0.48)f(2)=7.68X=-4.561d6.969600000 -(2,7.68)-0.96+- - 0.48(75)+4.8(7.5)10F(7.5) =9x=7.5x = 21(7.5,9) [1.7] SOLVE LINEAR-QUADRACTIC SYSTEMSEXAMPLE #2: DETERMINE WHETHER A LINEAR FUNCTIONS INTERSECTS A QUADRATIC FUNCTIONDetermine algebraically whether the given linear and quadratic functions intersect. If they dointersect, determine the number of points of intersectiona. f(x) = 3x + 5 & f(x) = 3x2 -2-24-4 -b. f(x)=-x-2&f(x)=-2x2+x-3 -3.2-21-4=3x+51=- 2I3x2-2x-3x-5-4=03x2-5x-9=01abc-Iabbs-62 4acI2 ROOTS62-4acNo ROOTS,= (-5)2-4(3)(9)I-22-4(-2)(-1)= 133:=-4EXAMPLE #3: DETERMINE THE Y-INTERCEPT FOR A TANGENT TO A QUADRATIC FUNCTIONIf a line with a slope of 4 has one point of intersection with the quadratic functionf(x)=1x2+2x-8, - what is the y-intercept of the line? Write the equation of the line in slopey-intercept form.1f(x) =(2.472, 0 (2.5,02F(x) = 4 x tk =(2,-2)5(0,-8)-10(-2,-10)(0,-10)tv?+2-8-k=ob2-4ac =02(2)2-4(=)(-8-K)=0c4-2(-8-k)=0Equation4+16+2K=02k=20f(x) = 4x-1022TK=10 [1.7] SOLVE LINEAR-QUADRACTIC SYSTEMSEXAMPLE #4: SOLVE A PROBLEM INVOLVING A LINEAR QUADRATIC SYSTEMDudley Do-Right is riding his horse at a top speed of 10 meters/second toward the bank, which is 100meters away when the bank robber begins to accelerate away from the bank going in the samedirection as Dudley. The robber's distance, d, in meters, away from the bank after 't' seconds can bemodelled by the equation d = 0.2t2a. Write a corresponding model for the position of Dudley Do-Right as a function of time.DudleyBank/10m/s dit) c 10t-100 d(t) = 0.2t2 Robberb. Will Dudley Do-Right catch the bank robber? If he does, find the time and position where thishappens. If not, explain why not.(36.18, 261.803)I10t-100= 0.2t22500=0.2t2-104+100 =200X=1502(0.2)100x = 10 + 1 200.450-(13.82 38.197)+ /(10,0)-50(0.0)50100x=36,18 x=13.822 d(13.82) = 10(13.82) -1003d (36.18) 2:10(36.+8) -100=38= 261.80Dudley first catches therobber at 3 Em, thenDudley re - cuptomes thethey escaperobber at 26/m3