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Warm Up - Quadratic Functions and Characteristics

This warm-up exercise covers graphing quadratic functions, identifying the vertex and axis of symmetry, and understanding the characteristics of quadratic equations. Students are also introduced to the intercept form of quadratic functions.

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Warm Up - Quadratic Functions and Characteristics

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  1. Warm Up (remember to do these on a separate piece of paper. I will check CH 1 and 2 Warm Ups next test) STATE THE VERTEX of each but only graph the first two! • y = 2x² 2) y= (x-3)² +1 3) y = x² +4 4) y = 3x² + 2 5) y = 2(x-3)² + 4 6) y = 2(x+3)² + 4 7) y = 2(x-3)² - 4

  2. Ch 2.2 Characteristics of Quadratic Functions

  3. The Standard Form of a parabola is f(x) = ax²+bx + c

  4. EX 1: Graph f(x) = 3x² -6x + 1. Label the vertex and axis of symmetry up Step 1: Identify the coefficients. Because a > 0, the parabola opens ______ Step 2: Find the vertex Step 3: Draw the axis of symmetry Step 4: Make a table centered around the x-coordinate of the vertex Another good point is the y-intercept c Step 5: Domain: ___________ Range: ____________ Inc: _______________ Dec: _______________ a=3 b = -6 c = 1

  5. Step 1: Identify the coefficients. Step 2: Find the vertex Step 3: Draw the axis of symmetry Step 4: Make a table centered around the x-coordinate of the vertex Another good point is the the y-intercept c You Try! a) f(x) = x²- 4x + 2 b) f(x) = 2x²- 4x + 2 Domain: ___________ Range: ____________ Inc: _______________ Dec: _______________ Domain: ___________ Range: ____________ Inc: _______________ Dec: _______________

  6. What are you wondering? https://www.youtube.com/watch?v=MsREqsNS_yI quadratic models can be used to answer many real-life questions!

  7. EX 2: Modeling with Quadratics An acrobat gets into a cannon and has been told his path will be modeled by where x is the distance from the cannon(in feet) and y is the height of his movement in the air above the ground (in feet). What is the highest point the acrobat will reach?

  8. You Try! b) A baker has modeled the monthly operating costs for making wedding cakes by the function y = 0.5x² -12x +150 where is the total cost in dollars and x is the number of cakes prepared. What is the minimum cost? How many cakes should be prepared to yield this minimum cost? a) The height h (in feet) of water spraying from a fire hose can be modeled by h(x) = −0.03x² + x + 25, where x is the horizontal distance (in feet) from the fire truck. What is the highest point the water reaches?

  9. When the graph of a quadratic function has at least one x-intercept, the function can be written in intercept form f(x) = a(x − p)(x − q), where a ≠ 0 What is “Intercept Form” of a quadratic function? • p and q are the x-intercepts • (p, 0) and (q, 0) • The line of symmetry goes through the vertex • The formula for the x-coordinate of the vertex is x = • To find the y-value of the vertex, plug the x-value into the equation as usual

  10. EX 3: How can we graph a quadratic function in Intercept Form? Graph f(x) = −2(x + 3)(x − 1). Label the x-intercepts, vertex, and axis of symmetry p = -3 and q = 1 down Step 1: Identify and plot the x-intercepts. Because a < 1, the parabola opens ______ Step 2: Find the vertex Step 3: Draw the axis of symmetry Step 4: Make a table centered around the vertex, Which includes the x-intercepts Domain: ___________ Range: ____________ Inc: _______________ Dec: _______________

  11. You Try! Graph f(x) = (x + 1)(x − 3) Label the x-intercepts, vertex, and axis of symmetry Graph f(x) = − (x -4)(x + 2). Label the x-intercepts, vertex, and axis of symmetry Domain: ___________ Range: ____________ Inc: _______________ Dec: _______________ Domain: ___________ Range: ____________ Inc: _______________ Dec: _______________

  12. EX 3: Modeling with Quadratics The parabola shows the path of your first golf shot, where x is the horizontal distance (in yards) and y is the corresponding height (in yards). The path of your second shot can be modeled by the function f(x) = −0.02x(x − 80). Which shot travels farther before hitting the ground? Which travels higher?

  13. You Try! The flight of your first golf shot can be modeled by the function y = -.001x(x-260) where x is the horizontal distance (in yards) and y is the corresponding height (in yards). Your second shot is modeled by the graph. Which shot travels farther before hitting the ground? Which travels higher?

  14. SUMmary! Essential Question: What types of symmetry does the graph of a quadratic function have? Describe this symmetry. Other Questions to consider: If you know the vertex of a parabola, can you graph the parabola? If you knew the vertex and one additional point on the graph, would that be enough to graph the parabola?

  15. HW 2.2 Characteristics of Quadratic Equations pg 61 #15-21all, 23, 25, 33-43odd, 49, 53-59odd, 65, 69, 71, 73

  16. Sorting Activity Instructions You are given 16 graphs and 32 equations. Each graph needs two equations pasted under it: vertex form and standard form. You might notice there are some duplicates – that’s because SOME equations are classified both as vertex AND standard form The easiest way to do this is to check the vertex and y-intercepts! Standard Form:

  17. HINT: Look for the “y” intercept” which is the value c in y= ax² + bx + cOR foil vertex form and simplify

  18. Standard Form Standard Form

  19. Y = -x² - 6x - 9

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