Mathematics Parabolas
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1 a place of mind F A C U L T Y O F E D U C A T I O N Department of Curriculum and Pedagog Mathematics Parabolas Science and Mathematics Education Research Group Supported b UBC Teaching and Learning Enhancement Fund
2 Parabolas F D
3 Parabolas I The graph of = x is shown in red. The graph is then shifted units right and 3 units down. Which one of the following equations represents the translated green graph? C. D. E. x x x x x
4 Solution Answer: D Justification: The graph of = x is shifted to the right units b replacing x with (x ). It is shifted down 3 units b replacing with ( + 3). x 3 x x 3 shifts graph, 3 The vertex of = ax is (0, 0). After shifting the graph, the vertex becomes (, 3). In general, the vertex of the equation a x p q is (p, q).
5 Parabolas II What is the vertex of the graph of 8 x 3 15? Is the vertex a maximum or minimum point? C. D. E. 8 15, , , , ,,,,, 15, maximum minimum maximum minimum maximum
6 Solution Answer: C Justification: Recall that the vertex of ax p q is (p, q). Therefore: 8 15 Vertex, 3 The vertex is a maximum because in a , 3 a x p q Since a x p is alwas negative in this case, the graph will open downwards. x
7 Parabolas III The red line shows the graph of = x. All the other lines are in the form: ax In which one of the graphs is 0 a 1? Green graph Blue graph C. Can graph D. Purple graph C. D.
8 Solution Answer: B 0 a 1 Justification: When, the graph of ax is alwas positive so it must lie above (or on) the x- axis. Therefore graph C and D are incorrect. Since a 1, the graph we are looking for must lie below the red graph because its -values are less than. x The correct answer is graph B: 1 x 3 C. D.
9 Parabolas IV Which one of the following is true about a and q if the equation of the given parabola is written in the form a x p q a x p q? C. D. E. a 0, a 0, a 0, a 0, a 0, q 0 q 0 q 0 q 0 q 0
10 Solution Answer: D Justification: The vertex of the graph is at (6, -4). This shows that 3 x 6 4 q 4 0 p 6 This point is a maximum point because the parabolas grows to negative infinit (it opens downwards). Therefore a 0
11 Parabolas V How man solutions are there to the equation 3 0 x 6 4? 3 x 6 4 No solutions 1 solution C. solutions D. Infinite solutions E. Cannot be determined
12 Solution Answer: A Justification: The parabola never crosses the x-axis, which means it has no zeroes. 3 x 6 4 Therefore 3 0 x 6 has no solutions. 4
13 Parabolas VI Which one of the graphs shown has 1 unique zero? D. Red graph Green graph C. Blue graph D. Purple graph E. Can graph C. E.
14 Solution Answer: B (Green) Justification: The vertex of the green graph is located at the point (-5,0). Onl when x = -5 does = 0, so it has 1 zero. Quadratics in the form ax p have 1 zero. The vertices of these quadratics lie on the x- axis at (p,0). C. D. E.
15 Parabolas VII Consider a parabola with a vertex at (, 5) and one x-intercept at., 0 7 What are the coordinates of the other x-intercept? C. D. E. 16,0 7 4, ,0,0, 0
16 Solution Answer: D Justification: The parabola is smmetric around the line x =., 5 The two x-intercepts should therefore be the same distance from the line x =. Distance from x = : Other x-intercept: , , 0
17 Parabolas VIII Consider a parabola with x-intercepts at (0,0) and (-6,0). What are the coordinates of its vertex? C. D. E. 3, 3 6, 0 3, 0 3, 3 Cannotbe determined 6, 0 0, 0
18 Solution Answer: E Justification: It is not possible to determine the -coordinate of the vertex onl knowing two x-intercepts. The answer was therefore Cannot be determined. It is possible to determine the x-value, however. The vertex must located at the midpoint between (-6,0) and (0,0), so its x-coordinate is -3. Both A and B were possible vertices.
19 Parabolas IX Consider a parabola with its vertex at (-3, -3) and one of its x- intercepts at (0, 0). Which one of the following equations represents this parabola? x 3 3 C. D. E. 3 x x x The parabolaisnot unique 3, 3 0, 0
20 Solution Answer: C Justification: The vertex is given so the equation is in the form: 0 ax 3 3 Plugging in the point (0,0) will then find the value for a. a a Note: When points are given and 1 is the vertex, an unique parabola can be found. 1 x , 3 0, 0
21 Parabolas X Consider the graph of the quadratic function: 4 x 3 What is the vertex of the parabola? C. D. E. 4, 3 4, 3 4, 3 4, 3 Noneof the above C. D.
22 Solution Answer: B Justification: Factor out (-1) from (-4-x): 4 x 3 (4 x) 3 This function is the same as 4 x 3, which has a vertex at (-4, 3). The extra negative does not affect the graph due to the square. (Tr expanding the function and completing the square again) C. D.
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Roots of Quadratic Equations Studio We ve discussed finding the vertex of a parabola. If we have a quadratic in the form y = a(x h) 2 + k, then the vertex is at the point (h,k), indeed the reason for writing
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Chapter 1 : BioMath: Transformation of Graphs Use the results in part (a) to identify the vertex of the parabola. c. Find a vertical line on your graph paper so that when you fold the paper, the left portion
) 2 + (y 2. x 1. y c x2 = y
Graphing Parabola Parabolas A parabola is a set of points P whose distance from a fixed point, called the focus, is equal to the perpendicular distance from P to a line, called the directrix. Since this
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Batman. Part 1 and 2. Sam wants to recreate the Batman symbol using graphs. Describe fully the brown, orange and blue graphs.
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