QUADRATIC FUNCTION REVIEW
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1 Name: Date: QUADRATIC FUNCTION REVIEW Linear and eponential functions are used throughout mathematics and science due to their simplicit and applicabilit. Quadratic functions comprise another ver important categor of functions. You studied these etensivel in Common Core Algebra I, but we will review man of their important characteristics in this unit. QUADRATIC FUNCTIONS An function of the form where the leading coefficient, a, is not zero. Eercise #1: Without the use of our calculator, evaluate each of the following quadratic functions for the specified input values. Recall that, according to the formal Order of Operations, eponent evaluation should alwas come first. f (b) g 5 (c) h 4 f 3 g h f 5 g 1 Graphs of quadratic functions form what are known as parabolas. The simplest quadratic function, and one that ou should be ver familiar with, is reviewed in the net eercise. Eercise #: Consider the simplest of all quadratic functions. Create a table of values to plot this function over the domain interval 3 3. h (b) Sketch a graph of this function on the grid to the right. (c) State the coordinates of the turning point of this parabola. (d) State the equation of this parabola s ais of smmetr. (e) Over what interval is this function increasing?
2 All quadratic functions that have unlimited domains (domains that consist of the set of all real numbers) have turning points and an ais of smmetr. Eercise #3: Consider the quadratic function f 6 5. Determine the turning point of this function. (b) What is the range of this quadratic? (c) Graph this function on the grid to the right. (d) Wh does this parabola open downward as opposed to which opened upward? (e) Between what two consecutive integers does the larger solution to the equation 65 0 lie? Show this point on our graph. Eercise #4: A sketch of the quadratic function intercepts and its turning point is shown below marked with points at its The -intercepts: A B (Zeroes) The -intercept: The turning point: D C A D B Over what interval is this function positive? C
3 Name: Date: FACTOR VS FACTORING 1. Factor using Greatest Common Factor (GCF.) (b) 4 40 (c) 6 45 (d) 18 9 (e) 8 (f) 6 7 (g) (h) Factor using the difference of perfect squares. 11 (b) 64 (c) 4 1 (d) Write each of the following trinomials in its factored form (b) 14 4 (c) (d) 5 6 (e) 5 6 (f) (g) 1 0 (h) Each of the following trinomials has a leading coefficient that is prime. Factor each (b) (c) 9 15 (d) Each of the following trinomials has a non-prime leading coefficient. Factor each (b)
4 FACTORING BY GROUPING COMMON CORE ALGEBRA II You now have essentiall three tpes of factoring: (1) greatest common factor, () difference of perfect squares, and (3) trinomials. We can combine gcf factoring with the other two to completel factor quadratic epressions. Toda we will introduce a new tpe of factoring known as factoring b grouping. This technique requires ou to see structure in epressions. Eercise #1: Factor a binomial common factor out of each of the following epressions. Write our final epression as the product of two binomials (b) 5 4 (c) (d) Some ver special polnomials can be factored b taking advantage of the structure we have seen in the last two problems. The ke is to do mindful manipulations of epressions so that the remain equivalent but are written as an overall product. When we factor b grouping we first etract common factors from pairs of binomials in four-term polnomials. If we are luck we are left with another binomial common factor. Eercise #4: Use the method of factoring b grouping to completel factor the following epressions. 3 (b) (c) (d)
5 THE ZERO PRODUCT LAW COMMON CORE ALGEBRA II One of the most important equation solving technique stems from a fact about the number zero that is not true of an other number: THE ZERO PRODUCT LAW If the product of multiple factors is equal to zero then at least one of the factors must be equal to zero. The law can immediatel be put to use in the first eercise. In this eercise, quadratic equations are given alread in factored form. Eercise #1: Solve each of the following equations for all value(s) of (b) (c) Eercise #: In Eercise #1(c), wh does the factor of 4 have no effect on the solution set of the equation? The Zero Product Law can be used to solve an quadratic equation that is factorable (not prime). To utilize this technique the problem solver must first set the equation equal to zero and then factor the non-zero side. Eercise #3: Solve each of the following quadratic equations using the Zero Product Law (b)
6 Eercise #4: Consider the sstem of equations shown below consisting of a parabola and a line and 4 5 Find the intersection points of these curves algebraicall. The Zero Product Law is etremel important in finding the zero s or -intercepts (zeroes) of a parabola. Eercise #5: The parabola shown at the right has the equation Write the coordinates of the two -intercepts of the graph. 3. (b) Find the -intercepts of this parabola algebraicall. Eercise #6: Algebraicall find the set of -intercepts (zeroes) for each parabola given below. (b) 4 1 (c)
7 COMPLETING THE SQUARE AND SHIFTING PARABOLAS Parabolas, and graphs more generall, can be moved horizontall and verticall b simple manipulations of their equations. This is known as shifting or translating a graph. You worked with this etensivel in Common Core Algebra I. The first eercise will review how to use a method known as completing the square to identif shifts and the turning point of a parabola. Eercise #1: The function equation is is shown alread graphed on the grid below. Consider the quadratic whose Using the method of completing the square, write this equation in the form h k. (b) Describe how the graph of would be shifted to produce the graph of (c) Sketch the graph of point (verte)? 8 18 b using its verte form in. What are the coordinates of its turning The algorithm of completing the square works best when a 1 and b is even in the form a b c.
8 But, it does work in ever case, even the mess ones. Eercise #3: Place each of the following quadratic functions in verte form and identif the turning point. 3 1 (b) 6 1 Eercise #4: The method of completing the square can be performed on the standard quadratic equation a b c and after much manipulation can be placed in the form: Based on this formula, what is the -coordinate of the turning point of an parabola? Be careful. b b a c a 4a (b) Use this formula to find the turning point of the parabola 10. (c) Verif our answer from part b placing the quadratic 10 into verte form. (d) Verif both answers b eamining a table on our calculator using the original equation. Eercise #5: Use the formula b to find the turning points for each of the following quadratic functions. a (b) f g 5 0 4
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