9-1 Quadratic Equations and Functions Warm Up Warm Up Lesson Presentation Lesson Presentation California Standards California StandardsPreview

9-1 Quadratic Equations and Functions Warm Up 1. Evaluate x 2 + 5x for x = 4 and x = –3. 2. Generate ordered pairs for the function y = x for x-values 36; – 6 – 2, – 1, 0, 1, and 2. x –2–2 –1–1 012 y

9-1 Quadratic Equations and Functions 21.0 Students graph quadratic functions and know that their roots are the x-intercepts. Also covered: 17.0 California Standards

9-1 Quadratic Equations and Functions quadratic equation quadratic function parabola minimum value maximum value vertex Vocabulary

9-1 Quadratic Equations and Functions Solutions of the equation y = x 2 are shown in the graph. Notice that the graph is not linear. The equation y = x 2 is a quadratic equation. A quadratic equation in two variables can be written in the form y = ax 2 + bx + c, where a, b, and c are real numbers and a ≠ 0. The equation y = x 2 can be written as y = 1x 2 + 0x + 0, where a = 1, b = 0, and c = 0.

9-1 Quadratic Equations and Functions Notice that the graph of y = x 2 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.

9-1 Quadratic Equations and Functions 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.

9-1 Quadratic Equations and Functions Additional Example 1A: Determining Whether a Point Is on a Graph Without graphing, tell whether each point is on the graph of (4, 16) 16 = 16 Substitute (4, 16) into Since (4, 16) is a solution of, (4, 16) is on the graph = ? = ?

9-1 Quadratic Equations and Functions Additional Example 1B: Determining Whether a Point Is on a Graph Without graphing, tell whether the function is on the graph of ( – 2, 10) 10 = 10 Substitute ( – 2, 10) into Since ( – 2, 10) is a solution of, ( – 2, 10) is on the graph. = ? = ?

9-1 Quadratic Equations and Functions Additional Example 1C: Determining Whether a Point Is on a Graph Without graphing, tell whether each point is on the graph of ( – 4, 0) Substitute ( – 4, 0) into Since ( – 4, 0) is not a solution of, ( – 4, 0) is not on the graph. = ? = ? 0 16  ≠

9-1 Quadratic Equations and Functions Check It Out! Example 1a Without graphing, tell whether the point is on the graph of x 2 + y = 2. (1, 1) 2 = 2 Substitute (1, 1) into x 2 + y = 2. x 2 + y = 2 Since (1, 1) is a solution of, (1, 1) is on the graph. x 2 + y = = ? = ?

9-1 Quadratic Equations and Functions Check It Out! Example 1b Without graphing, tell whether the point is on the graph of x 2 + y = 2. x 2 + y = 2 Substitute into x 2 + y = 2. Since is not a solution of x 2 + y = 2, is not on the graph.

9-1 Quadratic Equations and Functions Check It Out! Example 1c Without graphing, tell whether the point is on the graph of x 2 + y = 2. ( – 3.5, 10.5) ≠ 2 Substitute ( – 3.5, 10.5) into x 2 + y = 2 Since ( – 3.5, 10.5) is not a solution of ( – 3.5, 10.5) is not on the graph. x 2 + y =2,  ( – 3.5) = ? = ?

9-1 Quadratic Equations and Functions The graph of a quadratic function is a curve called a parabola. To graph a quadratic function, generate enough ordered pairs to see the shape of the parabola. Then connect the points with a smooth curve.

9-1 Quadratic Equations and Functions Additional Example 2A: Graphing Quadratic Functions Graph the quadratic function. xy –2 –1 – 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.

9-1 Quadratic Equations and Functions Additional Example 2B: Graphing Quadratic Functions Graph the quadratic function. y = – 4x 2 x –2 –2 –1 – y 0 –4–4 – 16 –4–4 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.

9-1 Quadratic Equations and Functions When choosing values of x, be sure to choose positive values, negative values, and 0. Helpful Hint

9-1 Quadratic Equations and Functions Graph each quadratic function. Check It Out! Example 2a y = x x –2 –2 –1 – y 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.

9-1 Quadratic Equations and Functions Graph the quadratic function. Check It Out! Example 2b y = – 3x x –2 –2 –1 – y 1 –2–2 – 11 –2–2 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.

9-1 Quadratic Equations and Functions As shown in the graphs in Examples 2A and 2B, some parabolas open upward and some open downward. Notice that the only difference between the two equations is the value of a. When a quadratic function is written in the form y = ax 2 + bx + c, the value of a determines the direction the parabola opens. A parabola opens upward when a > 0. A parabola opens downward when a < 0.

9-1 Quadratic Equations and Functions Additional Example 3A: Identifying the Direction of a Parabola Tell whether the graph of the quadratic function opens upward or downward. Explain. Since a > 0, the parabola opens upward. Identify the value of a. Write the function in the form y = ax 2 + bx + c by solving for y. Add to both sides.

9-1 Quadratic Equations and Functions Additional Example 3B: Identifying the Direction of a Parabola Tell whether the graph of the quadratic function opens upward or downward. Explain. y = 5x – 3x 2 y = – 3x 2 + 5x a = –3 Identify the value of a. Since a < 0, the parabola opens downward. Write the function in the form y = ax 2 + bx + c.

9-1 Quadratic Equations and Functions Check It Out! Example 3a Tell whether the graph of each quadratic function opens upward or downward. Explain. f(x) = –4x 2 – x + 1 Identify the value of a. a = –4 Since a < 0 the parabola opens downward.

9-1 Quadratic Equations and Functions Check It Out! Example 3b Tell whether the graph of each quadratic function opens upward or downward. Explain. y – 5x 2 = 2x – 6 Identify the value of a. a = 5 Since a > 0 the parabola opens upward. y – 5x 2 = 2x – 6 y = 5x 2 + 2x – 6 + 5x 2 Write the function in the form y = ax 2 + bx + c by solving for y. Add 5x 2 to both sides.

9-1 Quadratic Equations and Functions The minimum value of a function is the least possible y-value for that function. The maximum value of a function is the greatest possible y-value for that function. The highest or lowest point on a parabola is the vertex. Therefore, the minimum or maximum value of a quadratic function occurs at the vertex.

9-1 Quadratic Equations and Functions

9-1 Quadratic Equations and Functions Additional Example 4: Identifying the Vertex and the Minimum or Maximum Identify the vertex of each parabola. Then give the minimum or maximum value of the function. The vertex is (–3, 2), and the minimum is 2. The vertex is (2, 5), and the maximum is 5. A. B.

9-1 Quadratic Equations and Functions Check It Out! Example 4 Identify the vertex of each parabola. Then give the minimum or maximum value of the function. The vertex is (3, –1), and the minimum is –1. The vertex is (–2, 5) and the maximum is 5. a. b.

9-1 Quadratic Equations and Functions Unless a specific domain is given, the domain of a quadratic function is all real numbers. One way to find the range of a quadratic function is by looking at its graph. For the graph of y = x 2 – 4x + 5, the range begins at the minimum value of the function, where y = 1. All y-values greater than or equal to 1 appear somewhere on the graph. So the range is y  1.

9-1 Quadratic Equations and Functions Caution! You may not be able to see the entire graph, but that does not mean the graph stops. Remember that the arrows indicate that the graph continues.

9-1 Quadratic Equations and Functions Additional Example 5: Finding Domain and Range 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

9-1 Quadratic Equations and Functions Check It Out! Example 5a 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

9-1 Quadratic Equations and Functions Check It Out! Example 5b 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

9-1 Quadratic Equations and Functions Lesson Quiz: Part I 1. Without graphing, tell whether (3, 12) is on the graph of y = 2x 2 – Graph y = 1.5x 2. no

9-1 Quadratic Equations and Functions Lesson Quiz: Part II Use the graph for Problems Identify the vertex. 4. Does the function have a minimum or maximum? What is it? 5. Find the domain and range. D: all real numbers; R: y ≤ –4 maximum; – 4 (5, –4)