Quadratic Forms Formula Vertex Axis of Symmetry. 2. Write the equation in intercept form. 3. Identify the Vertex. 4. Identify the Axis of Symmetry.

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1 CC Algebra II Test # Quadratic Functions - Review **Formulas Name Quadratic Forms Formula Vertex Axis of Symmetry Vertex Form f (x) = a(x h) + k Standard Form f (x) = ax + b x + c x = b a Intercept Form f (x) = a(x p)(x q ) x = Vertical Axis of Symmetry Horizontal Axis of Symmetry y = 4p(x h) + k x = 4p(y k) + h y = k **Example. Write the equation in vertex form.. Write the equation in intercept form. 3. Identify the Vertex. 4. Identify the Axis of Symmetry. 5. Identify the Focus. 6. Identify the Directrix. 7. Identify the Domain and Range. 8. Describe where the function is increasing. 9. Describe where the function is decreasing.

2 **Practice Problems. Describe the transformation of each function from its parent function. Then identify the vertex. a. f (x) = (x + 3) + b. f (x) = 5x. Write a rule for g described by the transformation of the graph of f. Then identify the vertex. a. f (x) = x + ; translation unit left and units up b. f (x) = 4x + 5 ; horizontal stretch by a factor of and a translation units up, followed by a reflection in the x-axis 3. State the form in which each of the following equations is presented. Then find for each function: () the equation of its axis of symmetry, () its minimum or maximum value, (3) where the function is increasing and decreasing, and (4) its domain and range. f (x) = 3(x ) 4 g (x) = x + 6x + 3 h (x) = ( x 3 )(x + 7 ) 4. A passenger on a stranded lifeboat shoots a distress flare into the air. The height (in feet) of the flare above the water is given by f (t) = 6t(t 8 ), where t is time (in seconds) since the flare was shot. The passenger shoots a second flare, whose path is modeled in the graph shown. Justify your answers to the following: a. Which flare travels higher? b. Which remains in the air longer?

3 5. Given the function f (x) = x + 8 x + 9, state whether the vertex represents a maximum or minimum point for the function. Explain your answer. 6. Identify the focus, directrix, and axis of symmetry of each parabola. Graph the equation. y = 6x. 8 x = y **Regents Problems 7. Which equation represents a parabola with the focus at ( 0, ) and the directrix of y =? x = 8 y x = 4 y x = 8y x = 4y 8. Which equation represents a parabola with the focus at ( 0, 4) and the directrix of y =? y = x + 3 y = x + y = x + 3 y = 4 x A parabola has its focus at (,) and its directrix is y =. The equation of this parabola could be y = 8 (x + ) y = 8 (x + ) y = 8(x ) y = 8 (x )

4 0. Which equation represents the set of points equidistant from line l and point R shown on the graph below? (x + ) + (x + ) (x ) + (x ). The directrix of the parabola (y + 3 ) = ( x 4) has the equation y = 6. Find the coordinates of the focus of the parabola.

5 ANSWERS **Formulas Quadratic Forms Formula Vertex Axis of Symmetry Vertex Form f (x) = a(x h) + k Standard Form f (x) = ax + b x + c ( b a, f( b a ) ) x = b a ( Intercept Form f (x) = a(x p)(x q ), f( ) ) x = Vertical Axis of Symmetry Horizontal Axis of Symmetry **Example y = 4p(x h) + k Focus : ( h, k + p ) Directrix : y = k p x = 4p(y k) + h Focus : ( h + p, k) Directrix : p y = k. Write the equation in vertex form. y = a(x h) + k 0 = a( ) = a( 3) = 9 a = 9 a a = y = (x ) + 8. Write the equation in intercept form. y = a(x p)(x q ) 8 = a( ( ))( 4 ) 8 = a(3)( 3 ) 8 = 9 a = 8 y = ( x + )(x 4 ) 3. Identify the Vertex. (, 8) 4. Identify the Axis of Symmetry. x = 5. Identify the Focus. 4p = = 8 p p = 8 (, 8 8 ) 43 (, 8 ) 6. Identify the Directrix. y = y = 8

6 7. Identify the Domain and Range. Domain is all real numbers. Range is y 8 8. Describe where the function is increasing. Function is increasing to the left of x= 9. Describe where the function is decreasing. Function is increasing to the right of x=. a. Translation 3 units left, vertical stretch by a factor of, then translated units up; ( 3, ) b. Reflection in the x -axis and a vertical stretch by a factor of 5, then a translation unit down; (0, ). a. g ( x ) = ( x + ) + 3; (, 3) b. g ( x ) = x 7; (0, 7) 3. 3a. Vertex form () x = () min. = 4 (3) decr. x <, incr. x > (4) D : all real nos., R : y 4 3b. Standard form; () x = 4 () max. = 35 (3) incr. x < 4, decr. x > 4 (4) D : all real nos., R : y 35 3c. Intercept form () x = () min. = 5 (3) decr. x <, incr. x > (4) D : all real nos., R : y ab. First flare travels higher (56 ft) and remains in the air longer (8 sec). 5. The vertex represents a maximum since a < a. focus (0, 4), directrix y = 4, axis x = 0 6b. focus (, 0), directrix x =, axis of sym. y = 0 7. ANS: The vertex of the parabola is (0,0). The distance, p, between the vertex and the focus or the vertex and the directrix is. 8. ANS: 4 9. ANS: 4 The vertex is (,0) and p =. 0. ANS: 4 The vertex is (, -) and p =.. The vertex of the parabola is (4, -3). The x -coordinate of the focus and the vertex is the same. Since the distance from the vertex to the directrix is 3, the distance from the vertex to the focus is 3, so the y -coordinate of the focus is 0. The coordinates of the focus are (4,0).