Transformations with Quadratic Functions KEY

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1 Algebra Unit: 05 Lesson: 0 TRY THIS! Use a calculator to generate a table of values for the function y = ( x 3) + 4 y = ( x 3) x + y 4 Next, simplify the function by squaring, distributing, and collecting like terms. 1 1 y = (x 3) (x 3) + 4 Square (write twice) 6 y = (x + -3x + -3x + 9) + 4 Multiply (FOIL) 3 4 y = (x + -6x + 9) y = x + -1x Distribute 5 1 y = x 1x + Collect Like Terms 6 Would this expression generate the same table of values? YES Use your calculator to make sure. These equations are both examples of quadratic functions. More importantly, these equations show the two special forms of these types of functions: Form Equation Uses Standard Form y = ax + bx + c Simplified form Vertex Form y = a(x h) + k Transformations of parent function Sketch graphs Determine equation from a graph Transformation Effects: When a quadratic function is given in the vertex y = x form, the parent function undergoes the following transformations. y = a( x h) + k a The function reflects over the x-axis if a is negative. a The graph stretches or compresses by a factor of a > 1 If, the graph stretches vertically. 0 < a < 1 If, the graph compresses vertically.. 01, TESCCC 09/06/1 page 1 of 7

2 k h The vertex shifts vertically k units. If k is positive, it shifts up. If k is negative, it shifts down. The vertex shifts horizontally h units. If h is positive, (x (h)), it shifts right. If h is negative, (x (-h)), it shifts left. The vertex of the parabola is at coordinates (h, k) Algebra Unit: 05 Lesson: 0 Transformations with Quadratic Functions KEY Sample Problems From the quadratic parent function: A) State if there is a reflection over the x-axis. B) Identify any vertical shift. C) Identify any horizontal shift. D) Identify any vertical stretch or compression and by what factor. E) Determine the standard form of the quadratic equation. F) Sketch the graph of the quadratic parent function and the given function. 1) y = ( x ) + 4 ) y = ( x 4) 5 A) No reflection A) Reflection over x-axis B) Vertical shift up 4 B) Vertical shift down 5 C) Horizontal shift right C) Horizontal shift right 4 D) No vertical comp/stretch D) No vertical comp/stretch E) y = x 4x + 8 E) y = x + 8x 1 F) F) 3) f ( x) = ( x + 3) 4 4) 1 f ( x) = ( x + 5) A) No reflection A) No reflection B) Vertical shift down 4 B) No Vertical shift C) Horizontal shift left 3 C) Horizontal shift left 5 01, TESCCC 09/06/1 page of 7

3 D) Vertical stretch by D) Vertical comp by ½ E) f ( x) = x + 1x + 14 E) 1 5 f ( x) = x + 5x + Algebra Unit: 05 Lesson: 0 F) F) 01, TESCCC 09/06/1 page 3 of 7

4 Algebra Unit: 05 Lesson: 0 Use transformations and the vertex form to determine a function rule for the quadratic graph. 5) Reflects over x-axis Vertical shift up 5 Horizontal shift right 3 No vertical comp/stretch y = a(x h) + k y = -(x 3) + 5 6) No reflection Vertical shift down 4 Horizontal shift left 5 Vertical compression factor of ½ y = a(x h) + k y = ½ (x + 5) 4 01, TESCCC 09/06/1 page 4 of 7

5 Algebra Unit: 05 Lesson: 0 Practice Problems: From the quadratic parent function: A) State if there is a reflection over the x-axis. B) Identify any vertical shift. C) Identify any horizontal shift. D) Identify any vertical stretch or compression and by what factor. E) Determine the standard form of the quadratic equation. F) Sketch graph the quadratic parent function and the given function. 1) ( 5) y = x + ) f ( x) = ( x ) 4 A) No reflection A) No reflection B) Vertical shift down B) Vertical shift down 4 C) Horizontal shift left 5 C) Horizontal shift right D) No vertical comp/stretch D) No vertical comp/stretch E) y = x + 10x + 3 E) f ( x) = x 4x F) F) 3) f ( x) = 3( x ) + 1 4) 1 y = ( x 4) + 3 A) Reflection over x-axis A) No reflection B) Vertical shift up 1 B) Vertical shift up 3 C) Horizontal shift right C) Horizontal shift right 4 D) Vertical stretch by 3 D) Vertical comp by 1/3 E) f ( x) = 3x + 1x 11 E) y = x x , TESCCC 09/06/1 page 5 of 7

6 F) F) Algebra Unit: 05 Lesson: 0 01, TESCCC 09/06/1 page 6 of 7

7 Algebra Unit: 05 Lesson: 0 Use transformations and the vertex form to determine a function rule for the quadratic graph. 5) y = (x + 4) 6) y = - 0.5(x ) , TESCCC 09/06/1 page 7 of 7