Family of Quadratic Functions Lesson 4-2b

General Form Quadratic functions have the standard form y = ax2 + bx + c a, b, and c are constants a ≠ 0 (why?) Quadratic functions graph as a parabola

Zeros of the Quadratic Zeros are where the function crosses the x-axis Where y = 0 Consider possible numbers of zeros  None (or two complex) One Two

Axis of Symmetry Parabolas are symmetric about a vertical axis For y = ax2 + bx + c the axis of symmetry is at Given y = 3x2 + 8x What is the axis of symmetry?

Vertex of the Parabola The vertex is the “point” of the parabola The minimum value Can also be a maximum What is the x-value of the vertex? How can we find the y-value?

Vertex of the Parabola Given f(x) = x2 + 2x – 8 What is the x-value of the vertex? What is the y-value of the vertex? f(-1)= 1 – 2 – 8 = -9 The vertex is at (-1, -9)

Vertex of the Parabola Given f(x) = x2 + 2x – 8 Graph shows vertex at (-1, -9) Note calculator’s ability to find vertex (minimum or maximum)

Shifting and Stretching Start with f(x) = x2 Determine the results of transformations ___ f(x + a) = x2 + 2ax + a2 ___ f(x) + a = x2 + a ___ a * f(x) = ax2 ___ f(a*x) = a2x2 a) horizontal shift b) vertical stretch or squeeze c) horizontal stretch or d) vertical shift e) none of these

Experiment with Quadratic Function Spreadsheet Other Quadratic Forms Standard form y = ax2 + bx + c Vertex form y = a (x – h)2 + k Then (h,k) is the vertex Given f(x) = x2 + 2x – 8 Change to vertex form Hint, use completing the square Experiment with Quadratic Function Spreadsheet

Vertex Form Changing to vertex form Add something in to make a perfect square trinomial Changing to vertex form Subtract the same amount to keep it even. This gives us the ordered pair (h,k) Now create a binomial squared

Write the following in Vertex Form y = 𝒙 𝟐 +𝟖𝒙+𝟒

Vertex Form Changing to vertex form y = -3 𝒙 𝟐 + 6x - 1 Add something in to make a perfect square trinomial Changing to vertex form Factor a from the linear and the quadratic terms y = -3 𝒙 𝟐 + 6x - 1 y = -3 (𝒙 𝟐 - 2x + ) – 1+ y = -3 (𝒙 𝟐 - 2x + 1 ) – 1 + 3 Subtract the same amount to keep it even. y = -3 (𝒙−𝟏) 𝟐 – 4 This gives us the ordered pair (h,k) = (1, -4) Now factor to create a binomial squared

Write the following in Vertex Form y = −𝟒𝒙 𝟐 +𝟏𝟔𝒙 −𝟓

Write the following in Vertex Form y = 2 𝒙 𝟐 +𝟏𝟎𝒙+𝟕

For each function, the vertex of the function’s graph is given For each function, the vertex of the function’s graph is given. Find the unknown coefficients. y = -3 𝑥 2 +𝑏𝑥+𝑐, given (1,0)

y = 𝑐 - a 𝑥 2 - 2x, given vertex (-1,3) For each function, the vertex of the function’s graph is given. Find the unknown coefficients. y = 𝑐 - a 𝑥 2 - 2x, given vertex (-1,3)

Assignment Book Page 206: 26-31, 39-43odd, 44-47, 49-52