Quadratic Functions A quadratic function is described by an equation of the following form: ax² + bx + c, where a ≠ 0 The graphs of quadratic functions are __________ All parabolas have an axis of symmetry.

The point at which the axis of symmetry intersects a parabola is called the _________.

Consider the graph of y = ax² + bx + c The y-intercept is ___ The equation of the axis of symmetry is X = The x-coordinate of the vertex is The y-coordinate of the vertex of a quadratic function is the maximum or minimum value.

The graph of f(x)= ax² + bx + c –Opens up and had a minimum value when a > 0. Opens down and has a maximum value when a < 0. Ex:

Ways to Graph Quadratics You can ALWAYS make a table (by picking and plugging in numbers for x, solving for y, then graphing those ordered pairs). You will need about 5 – 6 points (sometimes more) to get a good picture of what the graph should look like. or Use the fact that parabolas are symmetrical so you don’t need so many points.

Example 1-1a Use a table of values to graph Graph these ordered pairs and connect them with a smooth curve. Answer: xy –2 –

Example 1-1b Use a table of values to graph Answer: xy

Example 1-2a Graph these ordered pairs and connect them with a smooth curve. Answer: Use a table of values to graph xy –2 –

Example 1-2b Use a table of values to graph Answer: xy

Graph Using Vertex and Axis of Symmetry Step 1:Write the equation of the axis of symmetry using the formula (remember x = # is a vertical line through that number)

Graph Using Vertex and Axis of Symmetry Step 2:Find the coordinates of the vertex by plugging in the x value you just found and solving for y. Step 3:Decide if the vertex is a –Maximum and opens down because the x² term is negative or –Minimum and opens up because the x² term is positive

Graph Using Vertex and Axis of Symmetry Step 4:Graph the function Plot the vertex Graph the axis of symmetry vertical line Choose a value for x not on the axis of symmetry, plug it in and solve for y. Graph this point and then use the axis of symmetry to graph the “same” point on the other side of the axis. Repeat this once to get another two points, then sketch the parabola.

Example 1-3a Consider the graph of Write the equation of the axis of symmetry. In Equation for the axis of symmetry of a parabola and Answer: The equation of the axis of symmetry is

Example 1-3b Consider the graph of Find the coordinates of the vertex. Since the equation of the axis of symmetry is x = –2 and the vertex lies on the axis, the x -coordinate for the vertex is –2. Original equation Simplify. Add. Answer: The vertex is at (–2, 6).

Example 1-3c Identify the vertex as a maximum or minimum. Answer: Since the coefficient of the x 2 term is negative, the parabola opens downward and the vertex is a maximum point.

Example 1-3d Graph the function. You can use the symmetry of the parabola to help you draw its graph. On a coordinate plane, graph the vertex and the axis of symmetry. (–2, 6) Choose a value for x other than –2. For example, choose –1 and find the y -coordinate that satisfies the equation. Original equation Simplify.

Example 1-3e Graph the function. (–2, 6) Graph (–1, 4). (–1, 4) Since the graph is symmetrical about its axis of symmetry x = –2, you can find another point on the other side of the axis of symmetry. The point at (–1, 4) is 1 unit to the right of the axis. Go 1 unit to the left of the axis and plot the point (–3, 4). (–3, 4)

Example 1-3f Graph the function. (–2, 6) Repeat this for several other points. (–1, 4) Then sketch the parabola. (–3, 4) (0, –2) (–4, –2)

Example 1-3g Consider the graph of a.Write the equation of the axis of symmetry. b.Find the coordinates of the vertex. c.Identify the vertex as a maximum or minimum. Answer:

Example 1-3h d.Graph the function. Answer: Consider the graph of

Example 1-4a Multiple-Choice Test Item Which is the graph of AB CD

Example 1-4b Read the Test Item You are given a quadratic function, and you are asked to choose the graph that corresponds to it. Solve the Test Item Find the axis of symmetry of the graph Equation for the axis of symmetry and

Example 1-4c The axis of symmetry is –1. Look at the graphs. Since only choices C and D have this as their axis of symmetry, you can eliminate choices A and B. Since the coefficient of the x 2 term is negative, the graph opens downward. Eliminate choice C. Answer: D

Example 1-4d Multiple-Choice Test Item Which is the graph of AB CD Answer:

Summary How can you always graph a quadratic equation? How does the axis of symmetry help us graph? When will you have a maximum vertex? Which way does it open? When will you have a minimum vertex? Which way does it open? Assignment: pg 528: 1-9, 11, 15