Chapter Algebra 1 Copyright Big Ideas Learning, LLC Worked-Out Solutions. Maintaining Mathematical Proficiency.

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1 Chapter Maintaining Mathematical Proficienc. The function q is of the form = f(x h), where h =. So, the graph of q is a horizontal translation units left of the. The function h is of the form = af(x), where a =.. So, the graph of h is a vertical shrink b a factor of. and a reflection in the x-axis of the. The function g is of the form = f(x h) + k, where h = and k =. So, the graph of g is horizontal translation units right and a vertical translation units up of the. The function p is of the form = af(x h), where a = and h =. So, the graph of p is a vertical stretch b a factor of and a horizontal translation unit left of the.. What Did You Learn? (p. 9). Sample answer: Because the highest point has a -coordinate of, the height is the opposite of the -coordinate of the lowest points. The width is the absolute value of the difference of the x-coordinates of the endpoints.. Sample answer: The t-intercept of the graph is the total time before the water balloon hits the ground.. Sample answer: Use the definition of vertex to identif f ( a) b as the -coordinate of the vertex. Then use the definition of maximum value/minimum value to recognize that this is also the maximum or minimum value of the function... Quiz (p. ). The vertex is (, ). The axis of smmetr is x =. The domain is all real numbers. The range is. When x <, increases as x increases. When x >, increases as x decreases.. The vertex is (, ). The axis of smmetr is x =. The domain is all real numbers. The range is. When x <, increases as x decreases. When x >, increases as x increases.. x h(x)... x p(x) p(x) = x + x The graph of p is a vertical stretch b a factor of and a vertical translation units up of the x r(x) 9 x r(x) = x The graph of r is a vertical stretch b a factor of and a vertical translation units down of the x b(x) b(x) = x x The graph of b is a vertical stretch b a factor of of the x h(x) = x The graph of h is a reflection in the x-axis of the Algebra Copright Big Ideas Learning, LLC

2 Chapter 7.. x g(x) g(x) = x x The graph of g is a vertical shrink b a factor of of the x m(x).. x m(x) = x The graph of m is a vertical shrink b a factor of, a reflection in the x-axis, and a vertical translation units down of the 9. The graph of g is a vertical translation units up of the graph of f. x f(x) = x g(x) = f(x) + g(x) = f(x) + g(x) = f(x) + f(x) = x + x g(x) = (x + ) + = x + ( + ) = x + So, g(x) = x +.. The graph of g is a vertical translation 9 units down of the x f(x) = x g(x) = f(x) g(x) = f(x) 9 f(x) = x + x g(x) = f(x) 9 g(x) = ( x + ) 9 = x + ( 9) = x + So, g(x) = x +.. The graph of g is a vertical translation units down of the x f(x) = x g(x) = f(x) f(x) = x g(x) = f(x) x g(x) = f(x) = ( x ) = x + ( ) = x So, g(x) = x. Copright Big Ideas Learning, LLC Algebra

3 Chapter. The graph of g is a vertical translation unit up of the graph of f. x.. f(x) = x.7.7 g(x) = f(x) g(x) = f(x) + g(x) = f(x) + g(x) = (x ) + x f(x) = x = x + ( + ) = x So, g(x) = x.. The axis of smmetr is a = () () = =. f(x) = x x + 7 f ( ) = ( ) ( ) + 7 = ( ) = = + 7 = So, the vertex is (, ). The -intercept is 7. So, the points (, 7) and (, 7) lie on the graph.. The axis of smmetr is a = () = =. f(x) = x + x + f( ) = ( ) + ( ) + = (9) + = + = + = So, the vertex is (, ). The -intercept is. So, the points (, ) and (, ) lie on the graph. 9 f(x) = x + x + x The domain is all real numbers. The range is.. The axis of smmetr is a = () = =. = x + x = ( ) + ( ) = = = 9 So, the vertex is (, 9). The -intercept is. So, the points (, ) and (, ) lie on the graph. x f(x) = x x + 7 = x + x x The domain is all real numbers. The range is 9. The domain is all real numbers. The range is. Algebra Copright Big Ideas Learning, LLC

4 Chapter. The axis of smmetr is a = ( ) = =. = x + x + 9 = () + () + 9 = () = = + 9 = So, the vertex is (, ). The -intercept is 9. So, the points (, 9) and (, 9) lie on the graph. = x + x + 9 x The domain is all real numbers. The range is. 7. For f(x) = x + x, a = and >, So, the parabola opens up, and the function has a minimum value. a = () = = f(x) = x + x f( ) = ( ) + ( ) = () = = = The minimum value is.. For f(x) = x + x +, a = and <. So, the parabola opens down, and the function has a maximum value. a = ( ) = = f(x) = x + x + f() = () + () + = () + + = + + = + = The maximum value is. 9. For = x + x +, a = and <. So, the parabola opens down, and the function has a maximum value. a = ( ) = = = x + x + = () + () + = + + = + = The maximum value is.. For = x + x +, a = and >. So, the parabola opens up, and the function has a minimum value. a = () = = = x + x +. = ( ) + ( ) + = () + = + = + = The minimum value is.. a. t.. (, ) = t t The point (, ) lies on the graph. So, it takes seconds for the coconut to fall feet. t = t + (., ). t The positive t-intercept is.. So, the pinecone hits the ground after. seconds. Copright Big Ideas Learning, LLC Algebra

5 Chapter b. = t + t.. b. x f(x) = x g(x) = (x ). = t + (., ). t The positive t-intercept of the graph that represents the height of the second pinecone is.. So, the second pinecone hits the ground after. seconds, which means the first pinecone hits the ground in the least amount of time.. t = b a = ( ) = = h(t) = t + t + h() = () + () + = () + + = + + = + = Because the midpoint of the graph occurs when x =, the domain is t. The highest point is the vertex (, ), and the lowest points are (, ) and (, ). So, the range is h. The maximum height of the softball is feet.. a. 7 f(x) = x g(x) = (x ) x Sample answer: The value of h causes a horizontal translation of the graph of = ax. x f(x) = x 9 g(x) = (x + ) 9 g(x) = (x + ) x f(x) = x. Explorations (p. ). a. x f(x) = x 9 g(x) = (x ) 9 f(x) = x g(x) = (x ) x b. x f(x) = x g(x) = (x + ) g(x) = (x + ) x f(x) = x Sample answer: The value of h causes a horizontal translation of the graph of = ax.. When h >, the graph of f(x) = a(x h) is a horizontal translation h units to the right of the graph of f(x) = ax. When h <, the graph of f(x) = a(x h) is a horizontal translation h units to the left of the graph of f(x) = ax. Algebra Copright Big Ideas Learning, LLC