1. Slides for 2/9 & 2/10 Precalculus

2. Warm-Up Set 9 Problem 2 Using your calculator, find a line of regression for the following data set, and find the correlation coefficient r : xy 06 29 416 618 820 1025 1231 1434

3. Objectives Today, we will: Graph a quadratic function by hand using transformations and using technology. Express a quadratic function in the appropriate form:  Standard form  Intercept form  Vertex form Find the intercepts (zeroes) of a quadratic function, and solve quadratic equations.

4. Graphing Quadratics Use your calculator to graph the five example quadratics given here. How are they alike? How are they different?

5. Standard Form for a Quadratic In the standard form of a quadratic, the shape of the parabola is controlled by the parameters a, b, and c. The y-intercept of the parabola is (0, c). If a is positive, the parabola opens upwards; if a is negative, the parabola opens downwards. The line of symmetry is x = -b/(2a). A quadratic must be in standard form to use the quadratic formula to find the x-intercepts.

6. Intercept Form for a Quadratic The parameter a in the intercept form does the same thing it did in the standard form. The x-intercepts of the parabola are (p, 0) and (q, 0) (watch your signs!). The y-intercept of the parabola is (0, a·p·q). The axis of symmetry is x = (p + q)/2. We factor a quadratic to put it in intercept form – but some quadratics cannot be factored! A quadratic function with no x- intercepts doesn’t have an intercept form.

7. Vertex Form for a Quadratic Again, a serves the same purpose in the vertex form as in the two previous forms. The line of symmetry is x = h. The vertex of the parabola is (h, k). The y-intercept of the parabola is (0, a·h² + k). We can find the vertex form from the standard form by completing the square.

8. Finding the Useful Parts When you graph a quadratic function, you must always list the following important parts:  The y-intercept  The x-intercepts (if it has them; some parabolas do not)  The line of symmetry  The vertex All three forms can give you the y-intercept and the line of symmetry. To find the x-intercepts, either change to intercept form, or change to standard form and use the quadratic formula. To find the vertex, either change to vertex form, or plug the x-value of the line of symmetry back into the quadratic function.

9. The Quadratic Formula Just in case you forgot, here’s the quadratic formula. It finds the x-values of the x-intercepts of the parabola. Since it requires a, b, and c, you have to be in the standard form to use the quadratic formula.