5-7 Warm Up – Discovery with Partners Graph f(x) = x2 - 3 and f(x) = x2 + 3 1. Make a table using -2 < x < 2. Instead of using y, replace with f(x). 2. What happened to the graphs? x f(x)= x2 + 3 f(x) f(x)= x2 - 3 -2 f(-2)= (-2)2 + 3 7 f(-2)= (-2)2 - 3 -1 f(-1)= (-1)2 + 3 4 f(-1)= (-1)2 - 3 f(0)= (0)2 + 3 3 f(0)= (0)2 - 3 1 f(1)= (1)2 + 3 f(1)= (1)2 - 3 2 f(2)= (2)2 + 3 f(2)= (2)2 - 3 x f(x) -2 -1 1 2

Using Transformations to Graph Quadratic Functions 5-7 SWBAT Transform quadratic functions. Describe the effects of changes in the coefficients of y = a(x – h)2 + k.

5-7 You can also graph quadratic functions by applying transformations to the parent function f(x) = x2. Transforming quadratic functions is similar to transforming linear functions (Lesson 2-6).

5-7 Recall that functions can also be reflected, stretched, or compressed.

5-7 Using the graph of f(x) = x2 as a guide, describe the transformations and then sketch a rough graph of each function using all different colors. State the vertex. g(x) = x2 – 5 h(x) = (x – 5)2 j(x) = (x + 5)2 k(x) = x2 + 5

5-7 Use the graph of f(x) = x2 as a guide, describe the transformations and then graph each function. State the vertex, domain and range. g(x) = (x – 2)2 + 4

5-7 Use the graph of f(x) = x2 as a guide, describe the transformations and then graph each function. State the vertex, domain and range g(x) = (x + 2)2 – 3

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5-7 Using the graph of f(x) = x2 as a guide, describe the transformations and then graph each function. Use a table to graph. Hint: PEMDAS 1 ( ) g x =- x 2 4 Because a is negative, g is a reflection of f across the x-axis. Because |a| = , g is a vertical compression of f by a factor of .

5-7 Using the graph of f(x) = x2 as a guide, describe the transformations and then graph each function. Use a table to graph. g(x) =2x2

5-7 If a parabola opens upward, it has a lowest point. If a parabola opens downward, it has a highest point. This lowest or highest point is the vertex of the parabola. The parent function f(x) = x2 has its vertex at the origin. You can identify the vertex of other quadratic functions by analyzing the function in vertex form. The vertex form of a quadratic function is f(x) = a(x – h)2 + k, where a, h, and k are constants.

af(bx - c) + d = a(x - h)2 + k 5-7 What happens to the graph… ‘h’ (a number is add/sub inside) |x+2| Add: moves left Sub: moves right ‘k’ (a number is add/sub outside) |x|+2 Add: moves up Sub: moves down ‘a’ Vertical stretch/shrink If negative, graph is reflected

5-7 HOMEWORK PG 290 #13, 14, 18 List all transformations, state domain/range/vertex, show parent function with a different color or a dashed line

5-7 f(x) = x – 3 f(x) = -5 f(x) = 2x + 1 f(x) = 2x f(x) = 3 f(x) = -3 Warm Up – Discovery: System of Equations For each system of equations shown in the table, determine the number of points of intersection. Mark one cell with an “x” for each row. System No Points of Intersection One point of Intersection Two points of Intersection f(x) = x – 3 f(x) = -5 f(x) = 2x + 1 f(x) = 2x f(x) = 3 f(x) = -3

Using Transformations to Graph Quadratic Functions 5-7 SWBAT Transform quadratic functions. Describe the effects of changes in the coefficients of y = a(x – h)2 + k.

5-7 Because the vertex is translated h horizontal units and k vertical from the origin, the vertex of the parabola is at (h, k). When the quadratic parent function f(x) = x2 is written in vertex form, y = a(x – h)2 + k, a = 1, h = 0, and k = 0. Helpful Hint

Writing Transformed Quadratic Functions 5-7 Writing Transformed Quadratic Functions Use the description to write the quadratic function in vertex form. The parent function f(x) = x2 is vertically stretched by a factor of and then translated 2 units left and 5 units down. g(x) = a(x – h)2 + k Vertical stretch by : 4 3 a = Translation 2 units left: h = –2 Translation 5 units down: k = –5 g(x) = (x + 2)2 – 5

5-7 Use the description to write the quadratic function in vertex form. The parent function f(x) = x2 is vertically compressed by a factor of and then translated 2 units right and 4 units down to create g. Vertical compression by : a = Translation 2 units right: h = 2 Translation 4 units down: k = –4 g(x) = a(x – h)2 + k g(x) = (x – 2)2 – 4

5-7 Use the description to write the quadratic function in vertex form. The parent function f(x) = x2 is reflected across the x-axis and translated 5 units left and 1 unit up to create g. Reflected across the x-axis: a is negative Translation 5 units left: h = –5 Translation 1 unit up: k = 1 g(x) = a(x – h)2 + k g(x) = –(x +5)2 + 1

5-7 For each system of equations shown in the table, determine the number of points of intersection. Mark one cell with an “x” for each row. System No Points of Intersection One point of Intersection Two points of Intersection f(x) = x2 f(x) = 3 f(x) = 0 f(x) = 2x + 1

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5-7 HOMEWORK PG 290 #21 – 24, 27 – 29, 54, 56