10.1: Quadratic Equations and Functions Do NOW 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions SECTION 10.2 Breaking Down a Quadratic 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Graph using a Table Pick five points and plug into the equation Connect the dots to make the shape Make sure to accurately label the zeros and the y-intercept, if applicable Make sure it is the correct shape 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions The Change in C For shifts of c of ax2 + bx + c = y, the change of the c, the y–intercept shifts up or down 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Example 1 Graph y = x2 x y –2 –1 1 2 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Example 1 Graph y = x2 x y –2 –1 1 2 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Example 2 Graph y = x2 + 2 and describe the transformation from the parent function. x y –2 –1 1 2 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions The Change in C For shifts of c of ax2 + bx + c = y, the change of the c, the y–intercept shifts up or down 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Your Turn Graph y = x2 – 3 and describe the transformation from the parent function. x y –2 –1 1 2 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Example 3 Graph y = 2x2 and describe the transformation from the parent function. x y –2 –1 1 2 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Example 3 Graph y = 2x2 and describe the transformation from the parent function. x y –2 –1 1 2 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions The Change in A For shifts of a of ax2 + bx + c = y, the bigger the a, the smaller the graph gets. 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Example 4 Graph y = (1/2)x2 x y –2 –1 1 2 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Example 5 Graph y = –(1/2)x2 x y –2 –1 1 2 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Your Turn Graph y = –x2 + 4 x y –2 –1 1 2 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Intro to Quadratics Quadratic Function is viewed as, ax2 + bx + c = 0, where a ≠ 0 Parabola is an u–shaped graph If a is positive, it opens up If a is negative, it opens down Vertex is the highest or lowest point of the graph Axis of Symmetry is the vertical line passing though the vertex (x of vertex) represented by a DOTTED–line 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Definitions Roots: The solutions to the equation Also known as solutions X-intercepts: Point(s) where the graph crosses the x–axis. Also known as roots and x-intercepts. Ordered Pair: (2, 0) Vertex: Minimum or maximum value Axis of Symmetry: Line that separates the graph in half; always is x equals 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Definitions Determine: How it opens: Does it open up or down? Roots: Where does it cross the x-axis? Y-Intercept: Where does it cross the y-axis? Vertex: The highest/lowest part of the graph Axis of Symmetry: Equation of which the X of the vertex Domain: The set of all x-values. Range: The set of all y-values. Minimum/Maximum: The highest or lowest part of the graph 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Worksheet Key –5 –5 < x < 4 –2 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Worksheet Key 7, 11.5} 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Definitions Quadratic Equation is viewed as, ax2 + bx + c = 0, where a ≠ 0 Parent Functions of Parabola: Change of a Change of c 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Example 5 Graph y = x2 – 9 and identify: Graph the function and label the vertex, roots, axis of symmetry, and y-intercept How it opens: Does it open up or down? Roots: Where does it cross the x-axis? Y-Intercept: Where does it cross the y-axis? Vertex: The highest/lowest part of the graph Line of Symmetry: Equation of which the X of the vertex Domain: The set of all x-values. Range: The set of all y-values. Minimum/Maximum: The highest or lowest part of the graph 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Example 5 Graph y = x2 – 9 and identify: x y –3 –2 –1 1 2 3 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Example 5 Graph y = x2 – 9 and identify: 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Example 5 Graph y = x2 – 9 and identify: How it opens 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Example 5 Graph y = x2 – 9 and identify: Roots: 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Example 5 Graph y = x2 – 9 and identify: Y-intercept 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

Example 5 Graph y = x2 – 9 and identify: Y-intercept (with a calculator) 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Example 5 Graph y = x2 – 9 and identify: The Vertex 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Example 5 Graph y = x2 – 9 and identify: The Vertex 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

Example 5 Graph y = x2 – 9 and identify: Axis of Symmetry Line of Symmetry is the X of the vertex. 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

Domain (as an inequality) 10.1: Quadratic Equations and Functions Example 5 Graph y = x2 – 9 and identify: Domain (as an inequality) 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

Domain (with a calculator) 10.1: Quadratic Equations and Functions Example 5 Graph y = x2 – 9 and identify: Domain (with a calculator) 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

Range (as an inequality) 10.1: Quadratic Equations and Functions Example 5 Graph y = x2 – 9 and identify: Range (as an inequality) 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Example 5 Graph y = x2 – 9 and identify: Minimum/Maximum 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Example 5 Graph y = x2 – 9 and identify: Axis of Symmetry x y –3 –2 –1 1 2 3 Roots Roots Y-intercept Vertex 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Example 6 Graph y = x2 – 4 and identify: Graph How it opens Roots Y-Intercept Vertex Line of Symmetry Domain Range Minimum/Maximum 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Your Turn Graph y = x2 – 1 and identify: Graph How it opens Roots Y-Intercept Vertex Line of Symmetry Domain Range Minimum/Maximum 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Example 7 Graph y = 2x2 – 8 and identify: Graph the function and label the vertex, roots, axis of symmetry, and y-intercept How it opens: Does it open up or down? Roots: Where does it cross the x-axis? Y-Intercept: Where does it cross the y-axis? Vertex: The highest/lowest part of the graph Line of Symmetry: Equation of which the X of the vertex Domain: The set of all x-values. Range: The set of all y-values. Minimum/Maximum: The highest or lowest part of the graph 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Example 7 Graph y = 2x2 – 8 and identify: x y –3 –2 –1 1 2 3 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Example 7 Graph y = 2x2 – 8 and identify: Graph How it opens Roots Y-Intercept Vertex Line of Symmetry Domain Range Minimum/Maximum 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Example 8 Graph y = (–1/2)x2 + 4 and identify: Graph the function and label the vertex, roots, axis of symmetry, and y-intercept How it opens: Does it open up or down? Y-Intercept: Where does it cross the y-axis? Vertex: The highest/lowest part of the graph Line of Symmetry: Equation of which the X of the vertex Domain: The set of all x-values. Range: The set of all y-values. Minimum/Maximum: The highest or lowest part of the graph 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Example 8 Graph y = (–1/2)x2 + 4 and identify: x y –3 –2 –1 1 2 3 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Example 8 Graph y = (–1/2)x2 + 4 and identify: 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Example 8 Graph y = (–1/2)x2 + 4 and identify: Graph How it opens Y-Intercept Vertex Line of Symmetry Domain Range Minimum/Maximum 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Your Turn Graph y = (1/2)x2 – 4 and identify: Graph the function and label the vertex, roots, axis of symmetry, and y-intercept {Domain: –4, –2, 0, 2, 4} How it opens: Does it open up or down? Y-Intercept: Where does it cross the y-axis? Vertex: The highest/lowest part of the graph Line of Symmetry: Equation of which the X of the vertex Domain: The set of all x-values. Range: The set of all y-values. Minimum/Maximum: The highest or lowest part of the graph 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Your Turn Graph y = (1/2)x2 – 4 and identify: x y –4 –2 2 4 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Your Turn Graph y = (1/2)x2 – 4 and identify: Graph How it opens Roots Y-Intercept Vertex Line of Symmetry Domain Range Minimum/Maximum 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Writing Equations Use the information given to create the equation from the original equation If it is shift on the y-intercept, it effects the c If it is shift on the size, it effects the a 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Example 9 Given the function y = x2 + 2, describe the shift in the vertex of the parabola if, in the function, 2 is changed to –2? 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Example 10 What is the equation of the function that would be shifted 11 units up from the graph of f(x) = x2 – 8 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Example 11 What is the equation of the function that would be shifted 11 units up from the graph of f(x) = x2 – 8? 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Your Turn What is the equation of the function that would be shifted 11 units down from the graph of f(x) = x2 + 3? 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions

10.1: Quadratic Equations and Functions Assignment Worksheet 2/24/2019 6:11 AM 10.1: Quadratic Equations and Functions