Standard Form of a Quadratic Function Lesson 4-2 Part 1 Algebra 2 Standard Form of a Quadratic Function Lesson 4-2 Part 1

Goals Goal Rubric To graph quadratic functions written in standard form. 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 Standard Form

Big Idea: Function and Equivalence Essential Question Big Idea: Function and Equivalence Why is the standard form of a quadratic function useful?

Graphing a Quadratic Function in Standard Form Two of the defining characteristics of a quadratic function are its vertex and axis of symmetry. Therefore, when graphing a quadratic function, the first thing you want to calculate are the vertex and the axis of symmetry. The vertex and axis of symmetry are related. The vertex is on the axis of symmetry and you can use the equation of the axis of symmetry to find the x-coordinate of the vertex.

Calculating the Properties of a quadratic function in standard form Axis of Symmetry The Vertex 3. The y-Intercept

Review Axis of Symmetry The axis of symmetry is the line through the vertex of a parabola that divides the parabola into two congruent halves.

Standard Form Calculating the Axis of Symmetry You can use a formula to calculate the axis of symmetry. The formula works for all quadratic functions.

Standard Form Calculating the Axis of Symmetry Procedure: Using the standard form of the quadratic function, y = ax2 + bx + c, find the values of a and b. Use the formula, . Write the axis of symmetry as an equation.

Example: Find the axis of symmetry of the graph of y = –3x2 + 10x + 9. Step 1. Find the values of a and b. Step 2. Use the formula. y = –3x2 + 10x + 9 a = –3, b = 10 Step 3. Write as an equation. The axis of symmetry is

Your Turn: Find the axis of symmetry of the graph of y = 2x2 + x + 3. Step 1. Find the values of a and b. Step 2. Use the formula. y = 2x2 + 1x + 3 a = 2, b = 1 Step 3. Write as an equation. The axis of symmetry is .

Example: Find the vertex. y = –3x2 + 6x – 7 Step 1 Find the x-coordinate of the vertex. a = –3, b = 10 Identify a and b. Substitute –3 for a and 6 for b. The x-coordinate of the vertex is 1.

Example: Continued The x-coordinate of the vertex is 1. y = –3x2 + 6x – 7 Step 2 Find the corresponding y-coordinate. y = –3x2 + 6x – 7 Use the function rule. = –3(1)2 + 6(1) – 7 Substitute 1 for x. = –3 + 6 – 7 = –4 Step 3 Write the ordered pair. The vertex is (1, –4).

Your Turn: The x-coordinate of the vertex is 2. Find the vertex. y = x2 – 4x – 10 Step 1 Find the x-coordinate of the vertex. a = 1, b = –4 Identify a and b. Substitute 1 for a and –4 for b. The x-coordinate of the vertex is 2.

Continued The x-coordinate of the vertex is 2. y = x2 – 4x – 10 Step 2 Find the corresponding y-coordinate. y = x2 – 4x – 10 Use the function rule. = (2)2 – 4(2) – 10 Substitute 2 for x. = 4 – 8 – 10 = –14 Step 3 Write the ordered pair. The vertex is (2, –14).

Standard Form Finding the y-intercept Recall that a y-intercept is the y-coordinate of the point where a graph intersects the y-axis. The x-coordinate of this point is always 0. For a quadratic function written in the form y = ax2 + bx + c, when x = 0, y = c. So the y-intercept of a quadratic function is (0, c).

Standard Form Graphing Quadratic Functions Procedure: Find the axis of symmetry, . Find the vertex, (h, k). Find the y-intercept. Find two more points on the same side of the axis of symmetry as the point containing the y-intercept. Graph the axis of symmetry, the vertex, the point containing the y-intercept, and two other points. Reflect the points across the axis of symmetry. Connect the points with a smooth curve. Confirm the direction the graph opens with the sign of a.

Example: Graph f(x) = 3x2 – 6x + 1. Step 1 Find the axis of symmetry. Use x = . Substitute 3 for a and –6 for b. = 1 Simplify. The axis of symmetry is x = 1. Step 2 Find the vertex. y = 3x2 – 6x + 1 The x-coordinate of the vertex is 1. Substitute 1 for x. = 3(1)2 – 6(1) + 1 = 3 – 6 + 1 Simplify. = –2 The y-coordinate is –2. The vertex is (1, –2).

Example: Continued f(x) = 3x2 – 6x + 1. Step 3 Find the y-intercept. y = 3x2 – 6x + 1 y = 3x2 – 6x + 1 Identify c. The y-intercept is 1; the graph passes through (0, 1).

Example: Continued f(x) = 3x2 – 6x + 1. Step 4 Find two more points on the same side of the axis of symmetry as the point containing the y-intercept. Since the axis of symmetry is x = 1, choose x-values less than 1. Substitute x-coordinates. Let x = –1. Let x = –2. y = 3(–1)2 – 6(–1) + 1 y = 3(–2)2 – 6(–2) + 1 = 3 + 6 + 1 = 12 + 12 + 1 Simplify. = 10 = 25 Two other points are (–1, 10) and (–2, 25).

Example: Continued f(x) = 3x2 – 6x + 1. Step 6 Reflect the points across the axis of symmetry. Connect the points with a smooth curve. Confirm a > 0, opens up. Step 5 Graph the axis of symmetry, the vertex, the point containing the y-intercept, and two other points. x = 1 (–2, 25) (–1, 10) (0, 1) (1, –2) x = 1 (–1, 10) (0, 1) (1, –2) (–2, 25)

Example: Graph the quadratic function. f(x) = 2x2 + 6x + 2 Step 1 Find the axis of symmetry. Use x = . Substitute 2 for a and 6 for b. Simplify. The axis of symmetry is x .

Continued f(x) = 2x2 + 6x + 2 Step 2 Find the vertex. y = 2x2 + 6x + 2 The x-coordinate of the vertex is . Substitute for x. = 4 – 9 + 2 Simplify. = –2 The y-coordinate is . The vertex is .

Continued f(x) = 2x2 + 6x + 2 Step 3 Find the y-intercept. y = 2x2 + 6x + 2 y = 2x2 + 6x + 2 Identify c. The y-intercept is 2; the graph passes through (0, 2).

Continued f(x) = 2x2 + 6x + 2 Step 4 Find two more points on the same side of the axis of symmetry as the point containing the y-intercept. Since the axis of symmetry is x = –1 , choose x values greater than –1 . Let x = –1 Let x = 1 y = 2(–1)2 + 6(–1) + 1 Substitute x-coordinates. y = 2(1)2 + 6(1) + 2 = 2 – 6 + 2 = 2 + 6 + 2 Simplify. = –2 = 10 Two other points are (–1, –2) and (1, 10).

Continued f(x) = 2x2 + 6x + 2 Step 6 Reflect the points across the axis of symmetry. Connect the points with a smooth curve. Confirm a > 0, opens up. Step 5 Graph the axis of symmetry, the vertex, the point containing the y-intercept, and two other points. (–1, –2) (1, 10) (–1, –2) (1, 10)

Your Turn: Graph the quadratic function. y + 6x = x2 + 9 f(x) = x2 – 6x + 9 Rewrite in standard form. Step 1 Find the axis of symmetry. Use x = . Substitute 1 for a and –6 for b. = 3 Simplify. The axis of symmetry is x = 3.

Continued f(x) = x2 – 6x + 9 Step 2 Find the vertex. y = x2 – 6x + 9 The x-coordinate of the vertex is 3. Substitute 3 for x. y = 32 – 6(3) + 9 = 9 – 18 + 9 Simplify. = 0 The y-coordinate is 0. . The vertex is (3, 0).

Continued f(x) = x2 – 6x + 9 Step 3 Find the y-intercept. y = x2 – 6x + 9 y = x2 – 6x + 9 Identify c. The y-intercept is 9; the graph passes through (0, 9).

Continued f(x) = x2 – 6x + 9 Step 4 Find two more points on the same side of the axis of symmetry as the point containing the y- intercept. Since the axis of symmetry is x = 3, choose x-values less than 3. Let x = 2 Let x = 1 y = 1(2)2 – 6(2) + 9 Substitute x-coordinates. y = 1(1)2 – 6(1) + 9 = 4 – 12 + 9 = 1 – 6 + 9 Simplify. = 1 = 4 Two other points are (2, 1) and (1, 4).

Continued f(x) = x2 – 6x + 9 Step 6 Reflect the points across the axis of symmetry. Connect the points with a smooth curve. Confirm a > 0, opens up. Step 5 Graph the axis of symmetry, the vertex, the point containing the y-intercept, and two other points. x = 3 (3, 0) (0, 9) (2, 1) (1, 4) (0, 9) (1, 4) (2, 1) x = 3 (3, 0)

Assignment Section 4-2 pt 1, Pg. 214 – 215; #6-18