Vertex and Standard Form Graphing
Warm Up – Translating Quadratics Discovery Activity
1. a. Graph the quadratic parent function 𝑦 = 𝑥 ! using the FreeGraCalc App. Insert a picture of it here: b. What is the vertex of the parent function? _______ 2. Graph each of the following using your FreeGraCalc App. Write the coordinates of the vertex (double tap graph and choose “Find minima and maxima” then tap near vertex graph). Indicate in which direction the graph moved compared to the parent function and by how many units. a. 𝑦 = (𝑥 + 3)! b. 𝑦 = (𝑥 − 7)!
c. 𝑦 = 𝑥 ! + 5 d. 𝑦 = 𝑥 ! − 2
e. 𝑦 = −𝑥 !
f. 𝑦 = (𝑥 − 4)! − 3 g. 𝑦 = −(𝑥 + 8)!
h. 𝑦 = (𝑥 + 1)! − 5
i. 𝑦 = −(𝑥 − 6)! + 4 3. Rewrite the parent function 𝑦 = 𝑥 ! using vertex form 𝑦 = (𝑥 + ℎ)! + 𝑘 such that the graph is translated. Then graph all 3 simultaneously on FreeGraCalc and insert the picture: a. Right 3, down 6 b. Flipped, left 3 c. Up 9, right 5 4. Summarize the effects of ℎ and 𝑘 within the form 𝑦 = (𝑥 + ℎ)! + 𝑘 on the graph of the parent function. What happens when ℎ is positive? What happens when ℎ is negative? What happens when 𝑘 is positive? What happens when 𝑘 is negative? Graphing Quadratic Function Basics
Standard Form
Vertex Form
𝑦 = 𝑎𝑥 ! + 𝑏𝑥 + 𝑐
𝑦 = 𝑎(𝑥 − ℎ)! + 𝑘
Axis of Symmetry: Vertex: Ex: 𝑦 = 2(𝑥 + 2)! − 5 Axis of Symmetry: Vertex: Ex: 𝑦 = −2𝑥 ! + 8𝑥 + 3 Key Facts
If a > 0, then the graph _____________________________________ If a < 0, then the graph _____________________________________ If a > 1 or if a < -­‐1, then the graph ____________________________ If -­‐1 < a < 1, then the graph _________________________________ The c-­‐value represents the _________________________________ Converting Between Forms
Vertex to Standard (FOIL or BOX method) 1. 𝑦 = (𝑥 − 1)! + 2 *** Standard to Vertex (Complete the square!) 1. 𝑦 = 𝑥 ! − 4𝑥 + 6 2. 𝑦 = 2(𝑥 + 2)! − 5 2. 𝑦 = 𝑥 ! − 6𝑥 + 17 Practice
Find the axis of symmetry and vertex of the following quadratic functions. 1. 𝑦 = (𝑥 + 3)! − 4 2. 𝑦 = −(𝑥 − 7)! + 10 3. 𝑦 = 2𝑥 ! − 6𝑥 + 3 4. 𝑦 = 3𝑥 ! − 4𝑥 − 2 Convert from vertex to standard form. 5. 𝑦 = (𝑥 − 5)! − 2 6. 𝑦 = −3(𝑥 + 1)! + 5 Convert from standard to vertex form. 7. 𝑦 = 𝑥 ! + 2𝑥 + 5 8. 𝑦 = 𝑥 ! + 8𝑥 − 1 9. 𝑦 = 𝑥 ! + 10𝑥 − 7 10. 𝑦 = 𝑥 ! + 4𝑥 − 9 11. 𝑦 = 𝑥 ! − 12𝑥 + 4 12. 𝑦 = 𝑥 ! − 2𝑥 − 1