Graphing Quadratic Functions in Vertex form Section 5.1 Graphing Quadratic Functions in Vertex form

Graphing Quadratic Functions When quadratics are graphed they look like a U or an upside down U if a > 0, If a < 0, The highest point or lowest point on the parabola is the _________. _____________is the line that runs through the vertex and through the middle of the parabola.

Parabolas Basic (parent function) of a parabola y = x2 Vertex will be at next two points will be at and

Example Graph f(x) = x2. Note that a = 1 in standard form. Which way does it open? What is the vertex? What is the axis of symmetry?

Graph Quadratic Functions of the Form f(x) = x2 + k. Objective 1 Graph Quadratic Functions of the Form f(x) = x2 + k.

Graphing Quadratic Functions Graphing the Parabola Defined by f(x) = x2 + k If k is positive, the graph of f(x) = x2 + k is the graph of y = x2 shifted . If k is negative, the graph of f(x) = x2 + k is the graph of y = x2 shifted .

Example Graph f(x) = x2. Note that a = 1 in standard form. Which way does it open? What is the vertex? What is the axis of symmetry? Graph g(x) = x2 + 3 and h(x) = x2 – 3. What is the vertex of each function? What is the axis of symmetry of each function?

Example (cont) x y f(x) = x2 g(x) = x2 + 3 h(x) = x2 – 3

Graph Quadratic Functions of the Form f(x) = (x – h)2 . Objective 2 Graph Quadratic Functions of the Form f(x) = (x – h)2 .

Graphing the Parabola Defined by f(x) = (x – h)2 If h is positive, the graph of f(x) = (x – h)2 is the graph of y = x2 shifted to the . If h is negative, the graph of f(x) = (x – h)2 is the graph of y = x2 shifted to the .

Example Graph f(x) = x2. Graph g(x) = (x – 3)2 and h(x) = (x + 3)2. What is the vertex of each function? What is the axis of symmetry of each function? Continued

Example (cont) f(x) = x2 g(x) = (x – 3)2 h(x) = (x + 3)2

Graph Quadratic Functions of the Form f(x) = (x – h)2 + k. Objective 3 Graph Quadratic Functions of the Form f(x) = (x – h)2 + k.

Graphing the Parabola Defined by f(x) = (x – h)2 + k The parabola has the same shape as y = x2. The vertex is (h, k), and the axis of symmetry is the vertical line x = h.

Example Graph g(x) = (x – 2)2 + 4. Continued

Example (cont) f(x) = x2 g(x) = (x – 2)2 + 4

Graph Quadratic Functions of the Form f(x) = ax2 . Objective 4 Graph Quadratic Functions of the Form f(x) = ax2 .

Graphing the Parabola Defined by f(x) = ax2 If a is positive, if a is negative,. If |a| > 1, the graph of the parabola is than the graph of y = x2. If |a| < 1, the graph of the parabola is than the graph of y = x2.

Example Graph f(x) = x2. Graph g(x) = 3x2 and h(x) = (1/3)x2. How do the shapes of the graphs compare? Continued

Example (cont) x y f(x) = x2 g(x) = 3x2 h(x) = (1/3)x2

Graph Quadratic Functions of the Form f(x) = a(x – h)2 + k. Objective 5 Graph Quadratic Functions of the Form f(x) = a(x – h)2 + k.

Example Graph g(x) = –4(x + 2)2 – 1. Find the vertex and axis of symmetry.