form y = ax" +bx+ c Part 2: Shifting and Scaling symmetry of is *= h / x = -2 Open the Desmos activity found at this link. The link is also provided on Canvas. You may choose "Continue without signing in." 9. Adjust the sliders for a, h, and k until the vertex of the parabola is (-2, 3) and the parabola's y-intercept is (0, 1). Write the equation of your parabola with the a, h, and k values substituted in. vertix ( h , kc ) f ( 2 ) = _ a ( x - h ) + K IC T V 10. Fill in the blanks for each parabola below. Try to do so without graphing them first, then verify your answers with the graph. (a) y = (z-5)2 +1 (c) y= 212 -3 2 ( * + 0 ) - 3 a = 1 h=5. - K =1 1 = 2 h =0 k=_-3 Vertex: ( ) Range: [2) Vertex: (0 , ] Range: (5, 1 ) Axis of symmetry: - *=4 x = 5 Axis of symmetry: (b) y = -(1+4)2 -2 (d) y= -3(x - 4)2 a = = L h = _- 4 k=_-2 a = - 3 h =4 k=_o Vertex: (h , Is) Range: Vertex: Range: ( - 4 , - 21 Axis of symmetry: Axis of symmetry: 11. Answer these questions about a generic parabola of the form f(x) = a(x - h) + k. Your answers will involve a, h, and/or k. (a) What are the coordinates of the parabola's vertex? (b) What is the equation of the parabola's axis of symmetry? (c) Under what condition does the parabola open upwards? In this case, what is the range of f? (d) Under what condition does the parabola open downwards? In this case, what is the range of f? (e) What is the domain of f? 1 1Part 3: Finding the r-intercepts 12. Move to the next slide in the Desmos activity. Adjust the sliders for a, h, and k until the vertex of the parabola is (4, -3) and one of the parabola's r-intercepts is (2, 0). Write the equation of your parabola with the a, h, and k values substituted in. f (z) = 13. What is the other r-intercept of the parabola from #12? 14. Suppose a parabola has vertex (-1, 2) and one of its x-intercepts is (3, 0). What are the coordinates of its other x-intercept? Explain how you know. 15. Use the Desmos activity to help you answer these questions about a generic parabola of the form f(x) = a(x - h)2 + h. Most of your answers should involve a, h, and/or k. (a) Under what condition does the parabola have exactly one x-intercept? In this case, what are its coordinates? (b) Under what condition does the parabola have no r-intercepts? Hint: Think about the signs of a, h, and/or k. (c) Under what condition does the parabola have two r-intercepts? 16. Start with the formula y = a(x - h) + k and algebraically solve for the x-intercepts. Your answer should involve a, h, and k. Hint: Which coordinate is always zero at an r-intercept? 17. Check to see if your formula in #16 is consistent with the observations you made in #15. 15Part 4: Standard Form The expression a(x - h) + k is known as vertex form. However, we usually encounter quadratic expressions in the form ax' + br + c, which is known as standard form. In Part 4, you will develop shortcuts to connect the two forms. 18. Start with the vertex form expression, a(x - h) + k, and fully multiply it out. 19. Use your expanded expression from #18 to fill in the blanks below. a(x - h)+k =. . 202 + . . x+ 20. Next, match each coefficient. You should already have a for the coefficient of a2. (a) Set your coefficient of a equal to b. This looks like b = (b) Set your constant term equal to c. This looks like c = 21. Solve your equation from #20(a) to isolate h. This looks like h = (* Important! *) Note: If you need the vertex, it is best to use this formula to find h, then find k by computing f(h). 22. Solve your equation from #20(b) to isolate k. This looks like k =. 23. Substitute your expressions for h and k from #21 and #22 into your formula for the x-intercepts from #16. Then, try to simplify it down to the traditional Quadratic Formula: x =- -6+ vb2 - 4ac 2a 16