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Chapter 6: Quadratic Functions Math 2201 Handout #2 Standard Form β’ β’ Standard form for the equation of a quadratic function is π¦ = ππ₯ 2 + ππ₯ + π where a, b, and c are real numbers and a cannot be equal to zero It can also be written using function notation π(π₯) = ππ₯ 2 + ππ₯ + π Determining the Properties of the Function Written in Standard Form β’ All properties are determined based on the information in handout #1 1. Vertex** β’ The vertex of a quadratic function is the turning point of the graph. β’ Determine the x-coordinate: π = β π ππ β’ Determine the y-coordinate: substitute the x-coordinate into the equation and solve for y 2. Direction of Opening β’ β’ β’ β’ 3. A quadratic function can open Up or Down It is determined by the sign of the leading coefficient (the value of βaβ) If βaβ is POSITIVE ( a > 0 ) then the graph opens UP If βaβ is NEGATIVE ( a < 0 ) then the graph opens DOWN Maximum or Minimum Point β’ The vertex is always the maximum or minimum point on the graph of a quadratic function. It is one OR the other. It canβt be both β’ The vertex is the MINIMUM point (Lowest) point on the graph when the graph OPENS UP. It is on the bottom of the graph. This happens when βaβ is positive ( a > 0 ) β’ The vertex is the MAXIMUM point (Highest) point on the graph when the graph OPENS DOWN. It is at the top of the graph. This happens when βaβ is negative ( a < 0 ) 4. Maximum or Minimum Value β’ The y-coordinate of the vertex is the highest or lowest value for y for the function β’ It is the MINIMUM value when the vertex is at the bottom of the graph, when the graph OPENS UP. This happens when βaβ is positive ( a > 0 ) β’ It is the MAXIMUM value when the vertex is at the top of the graph, when the graph OPENS DOWN. This happens when βaβ is negative ( a < 0 ) 5. Width of the Graph β’ The graph can be wider or thinner or the same width as the original graph of y = x2 . This is determined by the number value (ignore the sign) of the leading coefficient, βaβ. β’ The graph is THINNER than y = x2 when the value of βaβ is higher than 1 (a > 1) β’ The graph is THE SAME width as y = x2 when the value of βaβ is equal to 1 (a = 1) β’ The graph is WIDER than y = x2 when the value of βaβ is between 0 and 1 (0 < a < 1) 6. Y-intercept β’ The y-intercept of any graph is where the graph crosses the y-axis. It is the answer for y when the value of x is zero β’ Depending on the form of the equation, the y-intercept can be determined from the equation. In standard form the y-intercept is βcβ the constant number at the end 7. Axis (Line) of Symmetry β’ The line (or Axis) of symmetry for any parabola is the vertical line that is drawn down through the vertex. β’ The line of symmetry divides the graph into two equal parts that are mirror images of each other. Page 1 of 2 Chapter 6: Quadratic Functions Math 2201 β’ Has an equation π₯ = (π‘βπ π₯ β πππππππππ‘π ππ π‘βπ π£πππ‘ππ₯) 8. Handout #2 Domain β’ The domain is the x-values that are allowed to be used in the function. Any number can be put into the equation to get an answer for y. β’ The domain is All Real Numbers π₯ β π 9. Range β’ The range is the y-values that are possible based on the domain values. The range depends on (i) the direction of opening (determined by the sign of βaβ) and (ii) the ycoordinate of the vertex (k) β’ When the graph OPENS UP (a > 0) the range is the y-coordinate of the vertex and all the y-values above it. It is written y ο³ k where k is the y-coordinate of the vertex β’ When the graph OPENS DOWN (a < 0) the range is the y-coordinate of the vertex and all the y-values below it. It is written y ο£ k where k is the y-coordinate of the vertex 10. To Draw the Graph β’ β’ β’ β’ Determine and plot the coordinates of the vertex Draw the line of symmetry through the vertex Determine and plot the y-intercept; plot its mirror image across the line of symmetry Choose another x-value and find its y-value. Plot this point and its mirror image across the line of symmetry β’ If the vertex is on the y-axis then it is the y-intercept as well. To find a second pair of points, choose another x-value, find its y-coordinate. Plot this point and its mirror image (a) (b) π¦ = π₯ 2 β 8π₯ + 6 (c) π¦ = β2π₯ 2 + 4π₯ β 1 1 π¦ = 2 π₯ 2 + 2π₯ β 4 Direction of Opening Maximum/Minimum Point Width of Graph Vertex Axis of Symmetry Y-intercept Maximum/Minimum Value Domain Range y y 10 10 5 - 10 -5 -5 10 5 5 - 10 y 10 - x10 -5 5 5 -5 - 10 10 - x10 -5 5 -5 - 10 Page 2 of 2 10 x