Quadratic Graphs and Their Properties

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Presentation transcript:

Quadratic Graphs and Their Properties Section 9-1

Goals Goal Rubric To graph quadratic functions of the form y = ax2 and y = ax2 + c. Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary Quadratic Function Standard Form of a Quadratic Function Quadratic parent function Parabola Axis of symmetry Vertex Minimum Maximum

Quadratic Equation Solutions of the equation y = x2 are shown in the graph. Notice that the graph is not linear. The equation y = x2 is a quadratic equation. A quadratic equation in two variables can be written in the form y = ax2 + bx + c, where a, b, and c are real numbers and a ≠ 0. The equation y = x2 can be written as y = 1x2 + 0x + 0, where a = 1, b = 0, and c = 0.

Quadratic Equations and Their Graphs For any quadratic equation in two variables all points on its graph are solutions to the equation. all solutions to the equation appear on its graph.

Quadratic Function Notice that the graph of y = x2 represents a function because each domain value is paired with exactly one range value. A function represented by a quadratic equation is a quadratic function. the simplest quadratic function f(x) = x2 or y = x2 is the parent function.

Standard Form

Parabola The graph of a quadratic function is a curve called a parabola. A parabola is a U-shaped curve as shown at the right. To graph a quadratic function, generate enough ordered pairs to see the shape of the parabola. Then connect the points with a smooth curve.

Vertex The highest or lowest point on a parabola is the vertex. If a parabola opens upward, the vertex is the lowest point. If a parabola opens downward, the vertex is the highest point. Vertex is the highest point Vertex is the lowest point

Quadratic Functions of the Form f(x) = ax2 Graph the functions on one coordinate plane. x y 2 4 6 8 2 4 6 8 f(x) = x2 g(x) = –x2 x f(x)  2 4 1 1 2 x g(x)  2  4 1  1 1 2 f(x) = x2 g(x) = – x2 Notice that the graph of g(x) is a reflection of the graph of f(x) over the x-axis.

Quadratic Functions If a > 0 in y = ax2 + bx + c, the parabola opens upward. The vertex is a minimum point. If a < 0 in y = ax2 + bx + c, the parabola opens upward. The vertex is a maximum point. Vertex is the maximum Vertex is the minimum Opens up a > 0 Opens down a < 0

Minimums and Maximums

Example: The vertex is (–3, 2), and the minimum is 2. Identify the vertex of each parabola. Then give the minimum or maximum value of the function. A. B. The vertex is (–3, 2), and the minimum is 2. The vertex is (2, 5), and the maximum is 5.

Your Turn: The vertex is (–2, 5) and the maximum is 5. Identify the vertex of each parabola. Then give the minimum or maximum value of the function. a. b. The vertex is (–2, 5) and the maximum is 5. The vertex is (3, –1), and the minimum is –1.

Axis of Symmetry The vertical line that divides a parabola into two symmetrical halves is the axis of symmetry. The axis of symmetry always passes through the vertex of the parabola. Axis of symmetry Vertex Axis of symmetry Vertex

Axis of Symmetry & Zeros

Example: Find the axis of symmetry of each parabola. A. (–1, 0) Identify the x-coordinate of the vertex. The axis of symmetry is x = –1. B. Find the average of the zeros. The axis of symmetry is x = 2.5.

Your Turn: Find the axis of symmetry of each parabola. a. (–3, 0) Identify the x-coordinate of the vertex. The axis of symmetry is x = –3. b. Find the average of the zeros. The axis of symmetry is x = 1.

Graphing y = ax2 You can use a table of values to graph the quadratic. Also, use the fact that a parabola is symmetric. First, find the coordinates of the vertex and several points on one side of the vertex. Then reflect the points across the axis of symmetry. For graphs of functions of the form y = ax2, the vertex is at the origin. The axis of symmetry is the y-axis, or x = 0.

Example: Make a table of values. Choose values of x and Use a table of values to graph the quadratic function. Make a table of values. Choose values of x and use them to find values of y. x y 4 3 1 –2 –1 1 2 Graph the points. Then connect the points with a smooth curve.

Your Turn: Make a table of values. Choose values of x and Use a table of values to graph the quadratic function. y = –4x2 x –2 –1 1 2 y –4 –16 Make a table of values. Choose values of x and use them to find values of y. Graph the points. Then connect the points with a smooth curve.

Domain and Range Unless a specific domain is given, you may assume that the domain of a quadratic function is all real numbers. You can find the range of a quadratic function by looking at its graph. For the graph of y = x2 – 4x + 5, the range begins at the minimum value of the function, where y = 1. All the y-values of the function are greater than or equal to 1. So the range is y  1.

Example: Find the domain and range. Step 1 The graph opens downward, so identify the maximum. The vertex is (–5, –3), so the maximum is –3. Step 2 Find the domain and range. D: all real numbers R: y ≤ –3

Your Turn: Find the domain and range. Step 1 The graph opens upward, so identify the minimum. The vertex is (–2, –4), so the minimum is –4. Step 2 Find the domain and range. D: all real numbers R: y ≥ –4

Your Turn: Find the domain and range. Step 1 The graph opens downward, so identify the maximum. The vertex is (2, 3), so the maximum is 3. Step 2 Find the domain and range. D: all real numbers R: y ≤ 3

Width of the Parabola The coefficient of the x2 term in a quadratic function affects the width of a parabola. When |m| < |n|, the graph of y = mx2 is wider than the graph of y = nx2. As we learned earlier the sign of the coefficient of the x2 term tells the direction it opens. Positive it opens up and negative it opens down.

Example: Graph the functions on one coordinate plane. x y f(x) = x2 2 4 6 8 2 4 6 8 f(x) = x2 g(x) = 2x2 f(x) = x2 x g(x)  2 8 1 2 1 x f(x)  2 4 1 1 2 g(x) = 2x2 Notice that the graph of g(x) is narrower than the graph of f(x).

Example: Graph the functions on one coordinate plane. f(x) = x2 x y 2 4 6 8 2 4 6 8 g(x) = x2 f(x) = x2 x f(x)  2 4 1 1 2 1 2 1  2 g(x) x g(x) = x2 Notice that the graph of g(x) is wider than the graph of f(x).

Your Turn: Graph f(x) = x2 Graph g(x) = 3x2 and h(x) = (1/3)x2 How do the shapes of the graphs compare?

Solution: x y f(x) = x2 g(x) = 3x2 h(x) = (1/3)x2

Solution: Graph f(x) = x2 Graph g(x) = 3x2 and h(x) = (1/3)x2 How do the shapes of the graphs compare? The graph of g(x) is narrower than f(x) and the graph of h(x) is wider than f(x).

Properties of the Form f(x) = ax2 If a > 0, the graph of f(x) = ax2 will open upward. In addition, if 0 < a < 1, the opening in the graph will be “wider” than that of y = x2. If a > 1, the opening in the graph will be “narrower” then that of y = x2. If a < 0, the graph of f(x) = ax2 will open downward. In addition, if 0 < |a| < 1, the opening in the graph will be “wider” than that of y = x2. If |a| > 1, the opening in the graph will be “narrower” then that of y = x2. When |a| > 1, we say that the graph is vertically stretched by a factor of |a|. When 0 < |a| < 1, we say that the graph is vertically compressed by a factor of |a|.

Graphing y = ax2 + c The y-axis is the axis of symmetry for graphs of functions of the form y = ax2 + c. The value of c translates the graph up or down. When c is positive the curve shifts up. When c is negative the curve shifts down.

Quadratic Functions of the Form f(x) = x2 + c y 2 4 6 8 2 4 6 8 Graph the functions on one coordinate plane. g(x) = x2 + 3 f(x) = x2 g(x) = x2 + 3 f(x) = x2 x f(x)  2 4 1 1 2 x g(x)  2 7 1 4 3 1 2 Notice that the graph of g(x) is the graph of f(x) shifted 3 units upward.

Quadratic Functions of the Form f(x) = x2 – c Graph the functions on one coordinate plane. x y 2 4 6 8 2 4 6 8 f(x) = x2 g(x) = x2 – 2 f(x) = x2 x f(x)  2 4 1 1 2 x g(x)  2 2 1  1 1 g(x) = x2 – 2 Notice that the graph of g(x) is the graph of f(x) shifted 2 units downward.

Your Turn: Graph f(x) = x2 Note that a = 1 in standard form. Which way does it open? What is the vertex? What is the axis of symmetry? Graph g(x) = x2 + 3 and h(x) = x2 – 3 What is the vertex of each function? What is the axis of symmetry of each function?

Solution: x y f(x) = x2 g(x) = x2 + 3 h(x) = x2 – 3

Solution: Graph f(x) = x2 Note that a = 1 in standard form. Which way does it open? Up What is the vertex? (0, 0) What is the axis of symmetry? x = 0 Graph g(x) = x2 + 3 and h(x) = x2 – 3 What is the vertex of each function? (0, 3) & (0, -3) What is the axis of symmetry of each function? Both x = 0

Properties of the Form f(x) = ax2 + c Graphing the Parabola Defined by f(x) = ax2 + c If c is positive, the graph of f(x) = x2 + c is the graph of y = x2 shifted upward k units. If c is negative, the graph of f(x) = x2 + c is the graph of y = x2 shifted downward |k| units. The vertex is (0, c), and the axis of symmetry is the y-axis.

Assignment 9-1 Exercises Pg. 556 - 558: #8 – 46 even