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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 1. 2. 3. 4. following functions: 3x + 4y = 12 x – 2y = 8 y = |x| Now GRAPH #3 from above. 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. 7-5 Graphing y – k = a(x – h)2 Let’s make a table of values and then graph. yx 2 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. 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. 7-5 Graphing y – k = a(x – h)2 Now let’s examine how small changes affect what the graph looks like: How about y = -x2? 7-5 Graphing y – k = a(x – h)2 How about a slightly more generic y=ax2? y = 3x2 y = (1/2)x2 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. 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 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. 7-5 Graphing y – k = a(x – h)2 Next, we have y – k = ax2. y – 3 = 1x2 Y + 3 = 1x2 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). 7-5 Graphing y – k = a(x – h)2 Now put them all together. 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. 7-5 Graphing y – k = a(x – h)2 Examples 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. 7-5 Graphing y – k = a(x – h)2 Examples of intercepts… 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. 7-5 Graphing y – k = a(x – h)2 Graph y + 6 = 2(x + 1)2. Label the vertex, axis and find all intercepts. 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) 7-5 Graphing y – k = a(x – h)2 More examples? 7-5 Graphing y – k = a(x – h)2 HOMEWORK!!