 # Monday, 5/10Tuesday, 5/11Wednesday, 5/12Thursday, 5/13Friday, 5/14 Graphing & Properties of Quadratic Functions HW#1 Graphing & Properties of Quadratic.

## Presentation on theme: "Monday, 5/10Tuesday, 5/11Wednesday, 5/12Thursday, 5/13Friday, 5/14 Graphing & Properties of Quadratic Functions HW#1 Graphing & Properties of Quadratic."— Presentation transcript:

Terms of a quadratic y = ax 2 + bx + c Every quadratic has terms: Quadratic term: ax 2 Linear term: bx Constant term: c When the power of an equation is 2, then the function is called a quadratic a, b, and c are the coefficients Standard form of a quadratic

Graphs of Quadratics 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 -2 4 -1 1 0 0 1 1 2 2..... x y y = x 2

Finding the solutions of a quadratic 2. Find the values of x that make the equation equal to 0 1)Algebraically (last week and next slide to review) 2)Graphically (today  next slide) 1. Set y of f(x) equal to zero: 0 = ax 2 + bx + c In general equations have roots, Functions haves zeros, and Graphs of functions have x-intercepts

Directions: Find the zeros. Ex: f(x) = x 2 – 8x + 12 Factor and set y or f(x) = 0 (x – 2)(x – 6) = 0 x – 2 = 0 or x – 6 = 0 x = 2 orx = 6 Factors of 12 Sum of Factors, -8 1, 12 13 2, 6 8 3, 4 7 -1, -12 -13 -2, -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)

Key Concept: Quadratic Functions Parent Functionf(x) = x 2 Standard From f(x) = ax 2 + bx + c Type of GraphParabola Axis of Symmetry y-interceptc

Axis of symmetry examples http://www.mathwarehouse.com/geometry/ parabola/axis-of-symmetry.php

Vertex formula x = -b 2a 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 (, )

Example 1: HW Prob #11 Find the vertex: y = 4x 2 + 20x + 5 a = 4, b = 20 x = -b = -20 = -20 = -2.5 2a 2(4) 8 y = 4x 2 + 20x + 5 y = 4(-2.5) 2 + 20(-2.5) + 5 = -20 The vertex is at (-2.5,-20)

Example 2 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)

Example 3: HW Prob #14 Find the vertex: y = 5x 2 + 30x – 4 a = 5, b = 30 x = -b = -30 = -30 = -3 2a2(5) 10 y = 5x 2 + 30x – 4 y = 5(-3) 2 + 30(-3) – 4 = -49 The vertex is at (-3,-49)

Example 4 Find the vertex: y = 2(x-1) 2 + 7 Answer: (1, 7)

Example 5 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)

Example: y = x 2 – 4 (HW Prob #1) x y y = x 2 - 4 2. What is the vertex (, ) 4. What are the solutions: (x-intercepts) 3. What is the y-intercept: 1. What is the axis of symmetry? x y -2 0 -1 -3 0 -4 1 -3 2 0 (0, -4) x = -2 or x = 2 -4 x = 0

Example: y = -x 2 + 1 (HW Prob #3) x y y = -x 2 + 1 2. Vertex: (0,1) 3. x-intercepts: x = 1 or x = -1 4. y-intercept: 1 1. Axis of symmetry: x = 0 x y -2 -3 -1 0 0 1 1 0 2 -3