1.02E - Square Root Rule and Inverse | Lecture Note

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High School
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High School Physics
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2024
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1.02E FINDING RULE & INVERSE Square Root Function • Unless 𝑏 is given, you will use the Simplified Standard Form of the rule: 𝑓 𝑥 = 𝑎 ±(𝑥 − ℎ) + 𝑘 Note 𝑥, 𝑦 ℎ, 𝑘 • Determine if the point is to the left or right of the vertex • Left: 𝑥 < ℎ 𝑓 𝑥 = 𝑎 −(𝑥 − ℎ) + 𝑘 • Right: 𝑥 > ℎ 𝑓 𝑥 =𝑎 𝑥−ℎ+𝑘 Given Point and Vertex • Find the rule of the square root function that passes through 23, −4 with vertex −2, −5 1. 2. 3. 4. Determine Left/Right Sub in vertex Sub in point Solve for 𝑎 5. Summarize Ex 1 RIGHT 𝑏 = +1 𝑓 𝑥 =𝑎 𝑥+2−5 −4 = 𝑎 23 + 2 − 5 −4 = 𝑎 25 − 5 1=𝑎 5 1 =𝑎 5 1 𝑓 𝑥 = 𝑥+2−5 5 • Find the rule of the square root function that passes through −2, −4 with vertex 2, 2 1. 2. 3. 4. Determine Left/Right Sub in vertex Sub in point Solve for 𝑎 5. Summarize Ex 2 LEFT 𝑏 = −1 𝑓 𝑥 = 𝑎 −(𝑥 − 2) + 2 −4 = 𝑎 −(−2 − 2) + 2 −4 = 𝑎 4 + 2 −6 = 𝑎 2 −3 = 𝑎 𝑓 𝑥 = −3 − 𝑥 − 2 + 2 • A water slide has the shape of a semi-parabola based on a square root function. This scenario is depicted in the graph below, with the axes scaled in metres. A series of vertical beams spaced at 5m intervals support the slide. We want to find the length of each beam. vertex (5,12) and point (30,0) Ex 3 Vertex (5,12) and another point (30,0) 𝑓(𝑥) = 𝑎 𝑥 − 5 + 12 0 = 𝑎 30 − 5 + 12 −12 = 𝑎 25 −12 = 𝑎 5 𝑎 = −2.4 Ex 3 ∴ 𝑓 𝑥 = −2.4 𝑥 − 5 + 12 f(10) ≈ 6.63 m f(15) ≈ 4.41 m f(20) ≈ 2.70 m f(25) ≈ 1.27 m • Find the Rule: • • • Vertex (-1,1) Point (3,3) 𝑏=1 • 𝑓 𝑥 = 𝑎 𝑥+1+1 • 3= 𝑎 3+1+1 2=𝑎 4 2 = 2𝑎 𝑎=1 • 𝑓 𝑥 = 𝑥+1+1 Ex 4 • If given: • • • • Domain: Range: Extreme (max/min): Variation • • • • ℎ and sign of 𝑏 𝑘 and sign of 𝑎 𝑘 and sign of 𝑎 Combination of signs for 𝑎 and 𝑏 Key information Find the rule of the function with domain −∞, 5 that passes through −4, 0 and 1, −2 By domain: 𝑏 = −1, ℎ = 5 𝑓 𝑥 =𝑎 − 𝑥−5 +𝑘 Create two rules and solve system of equations: 0 = 𝑎 − −4 − 5 + 𝑘 0=𝑎 9+𝑘 0 = 3𝑎 + 𝑘 𝑘 = −3𝑎 𝑘 = −3 2 𝑘 = −6 Ex 5 −2 = 𝑎 − 1 − 5 + 𝑘 −2 = 𝑎 4 + 𝑘 −2 = 2𝑎 + 𝑘 −2 = 2𝑎 − 3𝑎 −2 = −𝑎 𝑎=2 𝑓 𝑥 =2 − 𝑥−5 −6 INVERSE Square Root Function • The inverse of the basic square root function is a function! 𝑟𝑎𝑛 𝑓 = 0, ∞ • 𝑓 𝑥 = 𝑥, 𝑥 ≥ 0 −1 • 𝑓 𝑥 ⇒ 𝑥 = 𝑦, 2 𝑥 = 𝑦 𝑑𝑜𝑚 𝑓 = 0, ∞ −1 2 𝑓 𝑥 = 𝑥 ,𝑥 ≥ 0 −1 Inverse of the Basic • The inverse of the basic square root function is a function! • 𝑓 𝑥 = 𝑥, 𝑥 ≥ 0 −1 • 𝑓 𝑥 ⇒ 𝑥 = 𝑦, 2 𝑥 =𝑦 −1 2 𝑓 𝑥 = 𝑥 ,𝑥 ≥ 0 𝑓 𝑥 = 𝑥, 𝑦 ≥ 0 𝑓 −1 2 𝑥 = 𝑥 , 𝑥 ≥ 0 Inverse of the Basic • The inverse of a square root function is a restricted quadratic function • HALF OF IT! INVERSE 1. 2. 3. 4. 5. Find the Range of the original function Switch the 𝑥 and the 𝑦 Isolate the y State the Domain 𝑟𝑎𝑛 𝑓 = 𝑑𝑜𝑚 𝑓 −1 Summarize Steps • 𝑓 𝑥 = −2 𝑥 − 3 + 2 𝑟𝑎𝑛 𝑓 = −∞, 2 • 𝑓 −1 𝑥 ⇒ 𝑦≤2 𝑥 = −2 𝑦 − 3 + 2 𝑥 − 2 = −2 𝑦 − 3 𝑥−2 − = 𝑦−3 2 𝑥−2 2 =𝑦−3 4 𝑓 −1 1 𝑥 = 𝑥−2 4 2 + 3; 𝑥 ≤ 2 Solve for Inverse Miss Connell Visions 1 •P41: 9a, 12 •P27: 7c Homework