Quadratic Functions A quadratic function is a function with a formula given by the standard form f(x) = ax2+bx+c, where a, b, c, are constants and Some quadratic functions can be expressed in factored form f(x) = a(x – r)(x – s), where a, r, and s are constants and The graph of a quadratic function is called a parabola. Conversion from standard form to factored form for a quadratic function is called factoring.

Finding the zeros of a Quadratic Function Input values x which satisfy f(x) = 0 are called zeros of f. Sometimes we can find the zeros of a quadratic by factoring. For example, if f(x) = x2 – x – 6, we have which becomes (x – 3)(x + 2) = 0, so the zeros are x = 3 and x = –2. (See Skills for Factoring, page 120, for more on factoring.) If we look for the zeros of f(x) = ax2 + bx + c, then by the quadratic formula we have: If b2 – 4ac < 0, then x1 and x2 are not real numbers and the graph does not cross the x-axis.

Modeling the height of a baseball A baseball is thrown straight up in the air. Suppose its height in feet is given by where t is in seconds. The graph of f is shown below. What are the zeros of f? How do they relate to the baseball?

Concavity and Quadratic Functions Since the graph of a quadratic f(x) is a parabola, we know from previous experience that it is either concave up for all x or it is concave down for all x. Another way to interpret the previous statement is that the average rate of change for a quadratic f(x) is either increasing for all x or decreasing for all x. See the figure below. slope negative slope positive slope zero

Vertex Form of a Quadratic Function A quadratic function can be defined by a formula in one of the following forms: • Standard form: y = ax2+bx+c, where a, b, c, are constants, • Vertex form: y = a(x–h)2+k, where a, h, k are constants, To convert a quadratic function from vertex form to standard form, simply multiply out the squared term. Conversion from standard form to vertex form is called completing the square. It is discussed on the next slide and on Page 125 of the textbook.

Example. Put the following quadratic function into vertex form by completing the square: First factor out the coefficient of x2, which is – 4.  Step 1: Divide the coefficient of x by 2, giving 3/2.  Step 2: Square the result: (3/2)2 = 9/4.  Step 3: Add the result after the x term, then subtract it.  Step 4: Factor the perfect square and simplify the rest. Finally, multiply through by the – 4. perfect square

The Vertex of a Parabola Recall that the graph of a quadratic is called a parabola. The parabola corresponding to y = a(x–h)2+k: • Has vertex (h, k). • Has axis of symmetry x = h. • Opens upward if a > 0 or downward if a < 0.

Finding the vertex of a parabola Example. For the previous example, graph the parabola and find the vertex. We note that the vertex is at (–3/2, 1). The graph follows:

Finding a formula for a parabola If we know the vertex of a quadratic function and one other point, we can use the vertex form to find its formula, as shown in the following example. Example. A parabola has vertex at (–3, 2) and (0, 5) is on the parabola. Find the formula for the corresponding quadratic, f(x). Use the vertex form with h = –3 and k =2. This results in To find the value of a, we substitute x = 0 and y = 5 into this formula, obtaining a = 1/3. The formula is therefore

Finding a formula for a parabola, continued. If three points on a parabola are given, we can use the standard form of the corresponding quadratic to find the formula. Example. Suppose the points (0, 6), (1, 0), and (3, 0) are on a parabola. Find a formula for the parabola. Use the standard form: y = ax2+bx+c. Since (0, 6) is on the parabola, it follows that c = 6. From the other two points, we have: This system can be solved simultaneously for a and b. We obtain a = 2 and b = –8. Thus, the equation of the parabola is y = 2x2–8x+6.

Relating the zeros of a quadratic function to factored form Recall that the zeros of a function f are values of x for which f(x) = 0. Recall that some quadratic functions f(x) can also be expressed in factored form: Example. Find the zeros of f(x) = x2–x –6. Set f(x) = 0 and solve for x. We have x2–x –6 = 0. We next express f(x) in factored form, so it will be easy to find the zeros. The zeros are x = 3 and x = –2. Note that these are the values r and s from the factored form.

Finding a formula for a parabola using the factored form Example. Suppose the points (0, 6), (1, 0), and (3, 0) are on a parabola, as in a previous example. Find a formula for the parabola using the factored form. Since the parabola has zeros at x = 1 and x = 3, its formula is Substituting x = 0, y = 6 gives 6 = 3a or a = 2. Thus, the equation is: If we multiply this out, we get y = 2x2–8x+6, which is the same result as before.

Two methods for finding the zeros of a quadratic The first method involves completing the square. Suppose we want the roots of x2 + 3x + 2 = 0. If we complete the square as before, we get (x + 1.5)2– 0.25 = 0. If we rewrite this as (x + 1.5)2 = 0.25, we can take the square root of both sides of the equation to get x + 1.5 = 0.5, which gives x = –1 and x = –2. The other method involves the use of the quadratic formula, which was presented on a previous slide. If we apply the quadratic formula to x2 + 3x + 2 = 0, we get This reduces to x = –1.5 0.5 . Again, x = –1 and x = –2.

What does it mean if a quadratic does not have real zeros What does it mean if a quadratic does not have real zeros? It means that the graph of the corresponding parabola does not cross the x-axis. Problem. If we have 4 feet of string, what is the rectangle of largest area which we can enclose with the string? Solution. If we let one side of the rectangle have length x, then the other side must have length (4–2x)/2. That is, the other side is 2–x. Therefore, the area of the rectangle is a(x) = x(2–x) = –x2+2x. If we write this in vertex form, we have a(x) = –(x–1)2+1. Thus, the vertex is at (1, 1), and the rectangle of maximum area is a square with a side length of 1 foot.

Summary for Quadratic Functions • The formulas for quadratic functions include the standard form, the vertex form, and (sometimes) the factored form. • To convert from standard form to vertex form we use a procedure known as completing the square. • The graph of a quadratic function is a parabola. The highest point (or lowest point) on the graph is called the vertex. • Quadratic functions have at most 2 zeros which can be found by factoring or by the quadratic formula. Their graphs are either concave up or concave down for all x.