<strong>Chapter</strong> 5Quadratic Functions5. First, enter the functions into the Y= menu. Then press GRAPH to view a comparisonof the three graphs.Note how the graph of g(x) = x 2 + 2 has the same shape as the graph of f(x) = x 2but it is shifted 2 units up; and the graph of h(x) = x 2 + 4 also has the same shape,but is shifted 4 units up.It thus appears that y = x 2 + k, for a positive k, has the same shape as f(x) = x 2 butis shifted k units up.7. The function f(x) = −5(x − 4) 2 − 5 is given in vertex form f(x) = a(x − h) 2 + k,where a = −5, h = 4, and k = −5. The vertex is (h, k) = (4, −5).9. The function f(x) = 3(x + 1) 2 is given in vertex form f(x) = a(x − h) 2 + k, wherea = 3, h = −1, and k = 0. The vertex is (h, k) = (−1, 0).11. The function f(x) = −7(x − 4) 2 + 6 is given in vertex form f(x) = a(x − h) 2 + k,where a = −7, h = 4, and k = 6. The vertex is (h, k) = (4, 6).13. The function f(x) = 1 ( )6 x +7 23 +38 is given in vertex form f(x) = a(x − h)2 + k,where a = 1/6, h = −7/3, and k = 3/8. The vertex is (h, k) = ( − 7 3 , 3 8).15. The function f(x) = −7(x − 3) 2 + 1 is given in vertex form f(x) = a(x − h) 2 + k,where a = −7, h = 3, and k = 1. The axis of symmetry is the vertical line through thevertex. h = 3, so the axis of symmetry is x = 3.17. The function f(x) = − 7 ( )8 x +1 24 +23 is given in vertex form f(x) = a(x−h)2 +k,where a = −7/8, h = −1/4, and k = 2/3. The axis of symmetry is the vertical linethrough the vertex. h = − 1 4 , so the axis of symmetry is x = − 1 4 .19. The function f(x) = − 2 ( )9 x +2 23 −45 is given in vertex form f(x) = a(x−h)2 +k,where a = −2/9, h = −2/3, and k = −4/5. The axis of symmetry is the vertical linethrough the vertex. h = − 2 3 , so the axis of symmetry is x = − 2 3 .21. The function f(x) = − 8 ( )7 x +2 29 +65 is given in vertex form f(x) = a(x−h)2 +k,where a = −8/7, h = −2/9, and k = 6/5. The axis of symmetry is the vertical linethrough the vertex. h = − 2 9 , so the axis of symmetry is x = − 2 9 .Version: Fall 2007
Section 5.1The Parabola23. First, sketch your coordinate system. Compare the quadratic function f(x) =(x + 2) 2 − 3 <strong>with</strong> f(x) = a(x − h) 2 + k and note that h = −2 and k = −3. Hence, thevertex is located at (h, k) = (−2, −3). The axis of symmetry is a vertical line throughthe vertex <strong>with</strong> equation x = −2. Make a table to find two points on either side of theaxis of symmetry. Plot them and mirror them across the axis of symmetry. Use all ofthis information to complete the graph of f(x) = (x + 2) 2 − 3.y10f(x)=(x+2) 2 −3x y = (x + 2) 2 − 3−1 −20 1(−2,−3)x10x=−2To find the domain of f, mentally project every point of the graph onto the x-axis, asshown on the left below. This covers the entire x-axis, so the domain= (−∞, ∞). Tofind the range, mentally project every point of the graph onto the y-axis, as shown onthe right below. The shaded interval on the y-axis is range= [−3, ∞).y10y10x10x10Version: Fall 2007