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1 Topic Graphs of Quadratic Functions Graphs of Quadratic Functions

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2 Topic Graphs of Quadratic Functions California Standard: 21.0 Students graph quadratic functions and know that their roots are the x -intercepts. What it means for you: You’ll learn about the shape of various quadratic graphs. Key words: quadratic parabola concave vertex line of symmetry root

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3 Topic Graphs of Quadratic Functions So far in this Chapter you’ve solved quadratic equations in several different ways. In this Section you’ll see how the graphs of quadratic functions can be plotted using the algebraic methods you’ve already seen.

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4 Topic The Graphs of Quadratic Functions are Parabolas Graphs of Quadratic Functions If you plot the graph of any quadratic function, you get a curve called a parabola. The graphs of y = ax 2 (for various values of a ) on the right show the basic shape of any quadratic graph. –6–4– y x –4 –6 y = x 2 ( a = 1) y = 3 x 2 ( a = 3) y = – x 2 ( a = –1) y = –½ x 2 ( a = –½)

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5 Topic Graphs of Quadratic Functions The parabola’s either a u-shaped or n-shaped curve depending on the sign of a. The graph of y = ax 2 is concave up (u-shaped — it opens upwards) when a > 0, but concave down (n-shaped — it opens downwards) when a < 0. –6–4– y x –4 –6 y = x 2 ( a = 1) y = 3 x 2 ( a = 3) y = – x 2 ( a = –1) y = –½ x 2 ( a = –½) u-shaped n-shaped

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6 Topic Graphs of Quadratic Functions All quadratic graphs have one vertex (maximum or minimum point). For the curves shown below, the vertex is at the origin (0, 0). All quadratic graphs have a vertical line of symmetry. For the graphs on the right, the line of symmetry is the y -axis. –6–4– y x –4 –6 y = x 2 ( a = 1) y = 3 x 2 ( a = 3) y = – x 2 ( a = –1) y = –½ x 2 ( a = –½) vertex line of symmetry A bigger value of | a | results in a steeper (narrower) parabola. For example, the graph of y = 3 x 2 is steeper than the graph of y = x 2.

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7 Topic Graphs of Quadratic Functions The basic shape of all quadratic graphs (that is, for any quadratic function y = ax 2 + bx + c ) is very similar to the ones you’ve just seen. However, the graph can be stretched or squashed, and in a different place relative to the x - and y -axes, depending on the exact values of a, b, and c. They’re all concave up or concave down depending on the sign of a (concave up if a > 0 and concave down if a < 0).

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8 Topic Guided Practice Solution follows… Graphs of Quadratic Functions Match the equations with their graphs below. 1. y = –3 x 2 2. y = x 2 – 2 3. y = 2 x y = – x 2 – 1 5. y = 2 x 2 D A C E B y –6–4– x –4 –6 8 –8 A B C DE

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9 Topic y = ax 2 + c is Like y = ax 2 but Moved Up or Down by c Graphs of Quadratic Functions This diagram shows the graphs of y = x 2 + c, for three values of c : The top and bottom parabolas in the diagram are both the same shape as the graph of y = x 2. (i) the graph of y = x is 1 unit higher up the y -axis. (ii) the graph of y = x 2 – 4 is 4 units lower down the y -axis. The only differences are: –6–4– y x –4 –6 y = x 2 – 4 y = x y = x 2

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10 Topic Graphs of Quadratic Functions The graph of y = x 2 – 4 crosses the x -axis when y = 0 — that is, when x 2 – 4 = 0 (or x = ±2). In fact, the x -intercepts of any quadratic graph y = ax 2 + bx + c are called the roots of the function, and they correspond to the solutions of the equation ax 2 + bx + c = 0. The graph of y = x does not cross the x -axis at all. This is because x = 0 does not have any real solutions. –6–4– y x –4 –6 y = x 2 – 4 y = x y = x 2 roots

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11 Topic Graphs of Quadratic Functions So the graph of a quadratic function may cross the x -axis twice ( y = x 2 – 4), may touch the x -axis in one place ( y = x 2 ), or may never cross it ( y = x 2 + 1). However, the graph will always have a y -intercept — the graph will always cross the y -axis at some point. It all depends on how many roots the quadratic function has.

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12 Topic Guided Practice Solution follows… Graphs of Quadratic Functions Describe the graphs of the quadratics below in relation to the graph of y = x y = x y = x 2 – 3 8. y = 2 x y = x 2 – y = – x y = –2 x 2 – y = x 2 shifted up 1 unit y = x 2 shifted down 3 units Narrower than y = x 2 and shifted up 2 units Wider than y = x 2 and shifted down 5 units y = x 2 reflected about the x –axis, shifted up 1 unit y = x 2 reflected about the x –axis, but narrower and shifted down 4 units

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y – x 13 Topic Guided Practice Solution follows… Graphs of Quadratic Functions The graphs in Exercises 12 and 13 are transformations of the graph of y = x 2. Find the equation of each graph y = 2 x y = –3 x y – x –4

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14 Topic Independent Practice Solution follows… Graphs of Quadratic Functions Match the equations with their graphs below. 1. y = x 2 – 1 2. y = – x 2 – 1 3. y = 3 x 2 4. y = – x 2 5. y = – x A E B C D 1 4 –6–4– y x –4 –6 AB C D E

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15 Topic Independent Practice Solution follows… Graphs of Quadratic Functions Describe the graphs of the quadratics below in relation to the graph of y = x y = x y = –4 x 2 8. y = –2 x y = x Wider than y = x 2 and shifted up 1 unit Reflected about the x –axis and narrower than y = x 2 Reflected about the x –axis, narrower than y = x 2, and shifted up 3 units Wider than y = x 2

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16 Topic Graphs of Quadratic Functions Round Up Now you know how the a and c parts of the equation y = ax 2 + c affect the graph. In the next Topic you’ll learn how to draw some quadratic graphs yourself.

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