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Algebra 2

Algebra 2. Chapter 7 Quadratic Equations and Functions. 7-5 Graphing y – k = a(x – h) 2. WARMUP: Create a table of (x, y) values for the following functions: 3x + 4y = 12 x – 2y = 8 y = |x| Now GRAPH #3 from above. 7-5 Graphing y – k = a(x – h) 2.

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Algebra 2

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  1. Algebra 2 Chapter 7 Quadratic Equations and Functions

  2. 7-5 Graphing y – k = a(x – h)2 • WARMUP: • Create a table of (x, y) values for the following functions: • 3x + 4y = 12 • x – 2y = 8 • y = |x| • Now GRAPH #3 from above.

  3. 7-5 Graphing y – k = a(x – h)2 • GOAL: To graph parabolas whose equations have the form y-k = a(x-h)2, and to find the vertices and the axes of symmetry.

  4. 7-5 Graphing y – k = a(x – h)2 • Let’s make a table of values and then graph.

  5. 7-5 Graphing y – k = a(x – h)2 • This resulting curve is called a parabola. • Notice that points (x, y) and (-x, y) are “mirror images” of each other across the y-axis. • Because of this, the y-axis is called the axis of symmetry, or just axis, of this parabola.

  6. 7-5 Graphing y – k = a(x – h)2 • The vertex of the parabola is where the parabola crosses its axis. • For y=x2, the vertex is the origin.

  7. 7-5 Graphing y – k = a(x – h)2 • Now let’s examine how small changes affect what the graph looks like: • How about y = -x2?

  8. 7-5 Graphing y – k = a(x – h)2 • How about a slightly more generic y=ax2? • y = 3x2 • y = (1/2)x2

  9. 7-5 Graphing y – k = a(x – h)2 • The graph of y=ax2 opens upward if a>0 and downward if a<0. • The larger |a| is, the “narrower” the graph is.

  10. 7-5 Graphing y – k = a(x – h)2 • Now what about y = a(x – h)2? • y = 1(x – 0)2 • y = 1(x – 3)2 • y = 1(x – (-3))2

  11. 7-5 Graphing y – k = a(x – h)2 • To graph y = a(x – h)2, slide the graph of y = ax2 horizontally h units. If h>0, slide to the right. If h<0, slide to the left. The graph has vertex (h, 0) and its axis is the line x=h.

  12. 7-5 Graphing y – k = a(x – h)2 • Next, we have y – k = ax2. • y – 3 = 1x2 • Y + 3 = 1x2

  13. 7-5 Graphing y – k = a(x – h)2 • To graph y – k = ax2, slide the graph of y = ax2 vertically k units. If k>0, slide it upward; if k<0, slide it downward. The graph has vertex (0, k) and its axis is the line x = 0 (the y-axis).

  14. 7-5 Graphing y – k = a(x – h)2 • Now put them all together.

  15. 7-5 Graphing y – k = a(x – h)2 • To graph y – k = a(x – h)2, slide the graph of y=ax2 horizontally h units and vertically k units. The graph has a vertex (h, k) and its axis is the line x = h.

  16. 7-5 Graphing y – k = a(x – h)2 • Examples

  17. 7-5 Graphing y – k = a(x – h)2 • The y-coordinate of a point where the graph crosses the y-axis is called the y-intercept. The x-coordinate of a point where a graph crosses the x-axis is called the x-intercept. • A parabola may have no x-intercepts, one x-intercept or two x-intercepts.

  18. 7-5 Graphing y – k = a(x – h)2 • Examples of intercepts…

  19. 7-5 Graphing y – k = a(x – h)2 • To find the y-intercept of a parabola, set x equal to zero and solve for y. • To find the x-intercepts of a parabola, set y equal to zero in the equation and solve the resulting quadratic equation for x. If the roots are real, they are the x-intercepts. If the roots are imaginary, then the graph has NO x-intercepts.

  20. 7-5 Graphing y – k = a(x – h)2 • Graph y + 6 = 2(x + 1)2. Label the vertex, axis and find all intercepts.

  21. 7-5 Graphing y – k = a(x – h)2 • Find an equation in y – k = a(x – h)2 form with: vertex is (4, 5) and contains (5, 3)

  22. 7-5 Graphing y – k = a(x – h)2 • More examples?

  23. 7-5 Graphing y – k = a(x – h)2 • HOMEWORK!!

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