SWBAT…analyze the characteristics of the graphs of quadratic functions Wed, 2/15 Agenda 1. WU (10 min) 2. Characteristics of quadratic equations (35 min) Warm-Up: Given the function f(x) = 3x 2 – 18x + 15, for what values of x does f(x) = 0. HW#9: Quadratics

Quadratic Functions and their Graphs

Standard form of a quadratic y = ax 2 + bx + c a, b, and c are the coefficients Example: y = 2x 2 – 3x + 10 a = 2 b = -3 c = 10 When the power of an equation is 2, then the function is called a quadratic function

Quadratic Functions and their Graphs The graph of any quadratic equation is a parabola To graph a quadratic, set up a table and plot points Example: y = x 2 x y x y y = x 2

Finding the solutions of a quadratic (Review) 1. Set the equation = 0 2. Set y or f(x) equal to zero: 0 = ax 2 + bx + c 3. Factor 4. Set each factor = 0 5. Solve for each variable 1)Algebraically (last week and next slide to review) 2)Graphically (today  in three slides) In general equations have roots, Functions haves zeros, and Graphs of functions have x-intercepts

Directions: Find the zeros of the below function. f(x) = x 2 – 8x = (x – 2)(x – 6) x – 2 = 0 or x – 6 = 0 x = 2 orx = 6 Factors of 12 Sum of Factors, -8 1, , 6 8 3, , , , -4 -7

Characteristics of Quadratic Functions The shape of a graph of a quadratic function is called a parabola. Parabolas are symmetric about a central line called the axis of symmetry. The axis of symmetry intersects a parabola at only one point, called the vertex. The lowest point on the graph is the minimum. The highest point on the graph is the maximum.  The maximum or minimum is the vertex

Axis of symmetry. x-intercept. vertex y-intercept x y Characteristics of Quadratic Functions To find the solutions graphically, look for the x-intercepts of the graph (Since these are the points where y = 0) maximum

Axis of symmetry examples parabola/axis-of-symmetry.php

Given the below information, graph the quadratic function. 1. Axis of symmetry: x = Vertex: (1.5, ) 3. Solutions: x = -1 or x = 4 4. y-intercept: (0, -5) (HW9 Prob #8)

x y... (0, -5) x = 4 x = -1 x = 1.5. (1.5, -6.25)

WU: From the HW graph: x y 2. What is the vertex: 4. What are the solutions: (x-intercepts) 5. What is the domain? 6. What is the range? 3. What is the y-intercept: 1. What is the axis of symmetry?

WU: Graph y = x 2 – 4 x y 2. What is the vertex: 4. What are the solutions: (x-intercepts) 5. What is the domain? 6. What is the range? 3. What is the y-intercept: 1. What is the axis of symmetry?

Ex: Graph y = x 2 – 4 x y y = x What is the vertex: 4. What are the solutions: (x-intercepts) 5. What is the domain? 6. What is the range? 3. What is the y-intercept: 1. What is the axis of symmetry? x y (0, -4) x = -2 or x = 2 (0, -4) x = 0

Ex: Graph y = -x x y y = -x Vertex: (0,1) 4. Solutions: x = 1 or x = y-intercept: (0, 1) 1. Axis of symmetry: x = 0 x y Domain: All real numbers 6. Range: y ≤ 1

Given the below information, graph the quadratic function. 1. Axis of symmetry: x = 1 2. Vertex: (1, 0) 3. Solutions: x = 1 (Double Root) 4. y-intercept: (0, 2) Hint: The axis of symmetry splits the parabola in half

x y. (1, 0) x = 1. (0, 2)

Finding the y-intercept Given y = ax 2 + bx + c, what letter represents the y-intercept. Answer: c

Calculating the Axis of Symmetry Algebraically Ex: Find the axis of symmetry of y = x 2 – 4x + 7 a = 1 b = -4 c = 7

Calculating the Vertex (x, y) Algebraically Ex1: Find the vertex of y = x 2 – 4x + 7 a = 1, b = -4, c = 7 y = x 2 – 4x + 7 y = (2) 2 – 4(2) + 7 = 3 The vertex is at (2, 3) Steps to solve for the vertex: Step 1: Solve for x using x = -b/2a Step 2: Substitute the x-value in the original function to find the y-value Step 3: Write the vertex as an ordered pair (, )

Ex: Graph y = -x 2 – 3 x y 2. What is the vertex: 4. What are the solutions: (x-intercepts) 5. What is the domain? 6. What is the range? 3. What is the y-intercept: 1. What is the axis of symmetry?

Ex2: (HW9 Prob #11) Find the vertex: y = 5x x – 4 a = 5, b = 30 x = -b = -30 = -30 = -3 2a2(5) 10 y = 5x x – 4 y = 5(-3) (-3) – 4 = -49 The vertex is at (-3, -49)

Example: Find the vertex of y = 4x x + 5 a = 4, b = 20, c = 5 y = 4x x + 5 y = 4(-2.5) (-2.5) + 5 = -20 The vertex is at (-2.5,-20) Steps to solve for the vertex: Step 1: Solve for x using x = -b/2a Step 2: Substitute the x-value in the original function to find the y-value Step 3: Write the vertex as an ordered pair (, ) Ex3: (HW5 Prob #9)

Ex4: Find the vertex: y = x 2 + 4x + 7 a = 1, b = 4 x = -b = -4 = -4 = -2 2a 2(1) 2 y = x 2 + 4x + 7 y = (-2) 2 + 4(-2) + 7 = 3 The vertex is at (-2,3)

Find the vertex: y = 2(x – 1) y = 2(x – 1)(x – 1) + 7 y = 2(x 2 – 2x + 1) + 7 y = 2x 2 – 4x y = 2x 2 – 4x + 9 a = 2, b = -4, c = 9 y = 7 Answer: (1, 7) (HW5 Prob #12)

Given y = x 2 + 6x + 8, find the following algebraically 1. Axis of symmetry 2. Vertex (as an ordered pair) 3. Solutions (x-intercepts) 4. y-intercept (as an ordered pair) 5. After finding the above, graph the function 6. Domain 7. Range