2.4 Linear-Quadratic Systems PDF

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Linear Quadratics lesson plan for EC....

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Date: ______________________

2.4 Solving Linear-Quadratic Systems Definition: To solve a linear-quadratic system means to find where a __________ and _________________ intersect. There are 3 possible solutions for a linear-quadratic system:

2 solutions

1 solutions

No solutions

Example 1: Find the point(s) of intersection of 𝑓(𝑥) = 𝑥 2 − 7𝑥 − 15 and 𝑔(𝑥) = 2𝑥 − 5 . Notes:

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Date: ______________________ 1

You try: Find the point(s) of intersection of 𝑓(𝑥) = 2 𝑥 2 + 2𝑥 − 8 and 𝑔(𝑥) = 4𝑥 − 10.

Example 2: How many times do 𝑓(𝑥) = 2𝑥 2 − 4𝑥 + 1 and 𝑔(𝑥) = 𝑥 − 6 intersect?

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Example 3: If a line with a slope of 4 has one point of intersection with the quadratic function 1 𝑓(𝑥) = 𝑥 2 + 2𝑥 − 8, what is the y-intercept of the line? 2

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2.4 Linear-Quadratic Systems Practice 1. Determine the point(s) of intersection of each linear-quadratic system algebraically. a) b) c) d) e)

𝑓(𝑥 ) = 𝑥 2 − 7𝑥 + 15 and 𝑔(𝑥 ) = 2𝑥 − 5 1 𝑓(𝑥) = 𝑥 2 + 2𝑥 − 8 and 𝑔(𝑥) = 4𝑥 − 10 2 𝑓(𝑥 ) = 3𝑥 2 − 16𝑥 + 37 and 𝑔(𝑥 ) = 8𝑥 + 1 1 𝑓(𝑥) = 𝑥 2 − 2𝑥 − 3 and 𝑔(𝑥) = −3𝑥 + 1 2 𝑓(𝑥 ) = −2𝑥 2 − 7𝑥 + 10 and 𝑔(𝑥 ) = −𝑥 + 2

2. Determine if each quadratic will intersect once, twice, or not at all with the given linear function. a) b) c) d)

𝑓(𝑥 ) = 2𝑥 2 − 4𝑥 + 1 and 𝑔(𝑥 ) = 𝑥 − 6 𝑓(𝑥 ) = 2𝑥 2 − 2𝑥 + 1 and 𝑔(𝑥 ) = 3𝑥 − 5 𝑓(𝑥 ) = −𝑥 2 + 3𝑥 − 5 and 𝑔(𝑥 ) = −𝑥 − 1 1 𝑓(𝑥) = 2 𝑥 2 + 4𝑥 − 2 and 𝑔(𝑥) = 𝑥 + 3 2

e) 𝑓(𝑥) = − 𝑥 2 + 𝑥 + 3 and 𝑔(𝑥) = 𝑥 3 3. Determine the value of the 𝑦-intercept of a line with the given slope that is tangent (one point of intersection) to the given curve. a) 𝑓(𝑥) = −2𝑥 2 + 5𝑥 + 4 and a line with a slope of 1 b) 𝑓(𝑥) = −𝑥 2 − 5𝑥 − 5 and a line with a slope of -3 4. A parachutist jumps from an airplane and immediately opens his parachute. His altitude, 𝑦, in metres, after 𝑡 seconds is modelled by the equation 𝑦 = −4𝑡 + 300. A second parachutist jumps 5 s later and freefalls for a few seconds. Her altitude, in metres, during this time, is modelled by the equation 𝑦 = −4.9(𝑡 − 5)2 + 300. When does she reach the same altitude as the first parachutist?

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Answers: 1. a) (5, 5) and (4, 3) b) (2, −2) c) (6, 49) and (2, 17) d) (−4, 13) and (2, −5) e) (−4, 6) and (1,1) 2. a) 0 solutions b) 0 solutions c) 1 solution d) 2 solutions e) 2 solutions 3. a) (0, 6) b) (0, −4) 4. 7.5 seconds...