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Slides for 2/9 & 2/10 Precalculus

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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

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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.

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

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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.

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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.

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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.

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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.

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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.

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