4Quadratic Function Properties Leading CoefficientThe graph of a quadratic function is a parabola—a U shaped graph that opens either upward or downward.A parabola opens upward if its leading coefficient a is positive and opens downward if a is negative.The highest point on a parabola that opens downward and the lowest point on a parabola that opens upward is called the vertex. (The graph of a parabola changes shape at the vertex.)The vertical line passing through the vertex is called the axis of symmetry.The leading coefficient a controls the width of the parabola. Larger values of |a| result in a narrower parabola, and smaller values of |a| result in a wider parabola.
6Quadratic Functions Vertex Vertex yxThe graph of a quadratic function is a parabola.VertexA parabola can open up or down.If the parabola opens up, the lowest point is called the vertex.If the parabola opens down, the vertex is the highest point.VertexNOTE: if the parabola opened left or right it would not be a function!
7Standard Form a > 0 a < 0 yxThe standard form of a quadratic function isa < 0a > 0y = ax2 + bx + cThe parabola will open up when the a value is positive.The parabola will open down when the a value is negative.
8The line of symmetry ALWAYS passes through the vertex. Parabolas have a symmetric property to them.If we drew a line down the middle of the parabola, we could fold the parabola in half.We call this line the line of symmetry.Or, if we graphed one side of the parabola, we could “fold” (or REFLECT) it over, the line of symmetry to graph the other side.The line of symmetry ALWAYS passes through the vertex.
9Finding the Line of Symmetry When a quadratic function is in standard formFor example…Find the line of symmetry of y = 3x2 – 18x + 7y = ax2 + bx + c,The equation of the line of symmetry isUsing the formula…This is best read as …the opposite of b divided by the quantity of 2 times a.Thus, the line of symmetry is x = 3.
10Finding the Vertex Therefore, the vertex is (2 , 5) y = –2x2 + 8x –3 But we know the line of symmetry always goes through the vertex.y = –2x2 + 8x –3STEP 1: Find the line of symmetryThus, the line of symmetry gives us the x – coordinate of the vertex.STEP 2: Plug the x – value into the original equation to find the y value.To find the y – coordinate of the vertex, we need to plug the x – value into the original equation.y = –2(2)2 + 8(2) –3y = –2(4)+ 8(2) –3y = –8+ 16 –3y = 5Therefore, the vertex is (2 , 5)
11ExampleUse the graph of the quadratic function shown to determine the sign of the leading coefficient, its vertex, and the equation of the axis of symmetry.Leading coefficient: The graph opens downward, so the leading coefficient a is negative.Vertex: The vertex is the highest point on the graph and is located at (1, 3).Axis of symmetry: Vertical line through the vertex with equation x = 1.
16Graphing Quadratic Function: Method 3 The standard form of a quadratic function is given byy = ax2 + bx + cSTEP 1: Find the line of symmetrySTEP 2: Find the vertexSTEP 3: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve.
17Thus the line of symmetry is x = 1 Graphing Quadratic Function: Method 3Let's Graph ONE! Try …y = 2x2 – 4x – 1yxSTEP 1: Find the line of symmetryThus the line of symmetry is x = 1
18Graphing Quadratic Function: Method 3 Let's Graph ONE! Try …y = 2x2 – 4x – 1yxSTEP 2: Find the vertexSince the x – value of the vertex is given by the line of symmetry, we need to plug in x = 1 to find the y – value of the vertex.Thus the vertex is (1 ,–3).
19Graphing Quadratic Function: Method 3 Let's Graph ONE! Try …y = 2x2 – 4x – 1yxSTEP 3: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve.32yx–15
20The quadratic function f(x) =ax2 + bx + c can be written in an alternate form that relies on the vertex (h, k).Example: f(x) = 3(x - 4)2 + 6 is in vertex form with vertex (h, k) = (4, 6).What is the vertex of the parabola given byf(x) = 7(x + 2)2 – 9 ?Vertex = (-2,-9)
21ExampleWrite the formula f(x) = x2 + 10x + 23 in vertex form by completing the square.Given formulaSubtract 23 from each side.Add (10/2)2 = 25 to both sides.Factor perfect square trinomial.Subtract 2 form both sides..What is the vertex?Vertex is h= -5 k = -2.
22Example: Convert the quadratic f(x) = 3(x + 2)2 – 8 which is in vertex form to the form f(x) = ax2 + b + cGiven formula.f(x) = 3(x + 2)2 – 8Expand the quantity squared.f(x) = 3(x2 + 4x + 4) - 8Multiply by the 3.f(x) = 3x2 + 12xSimplfy.f(x) = 3x2 + 12x + 4
23Part 2: Polynomial Functions Properties of Basic Polynomial Function
24Basic Polynomial Function NOTE: Linear and Quadratic functions we considered earlier belong to the family of polynomial functions!