5-Minute Check on Chapter 2 Transparency 3-1 Click the mouse button or press the Space Bar to display the answers. 1.Evaluate 42 - |x - 7| if x = -3 2.Find 4.1  (-0.5) Simplify each expression 3. 8(-2c + 5) + 9c 4. (36d – 18) / (-9) 5.A bag of lollipops has 10 red, 15 green, and 15 yellow lollipops. If one is chosen at random, what is the probability that it is not green? 6. Which of the following is a true statement Standardized Test Practice: ACBD 8/4 < 4/8-4/8 < -8/4-4/8 > -8/4-4/8 > 4/8

Lesson 10-2 Solving Quadratic Equations by Graphing

Transparency 2 Click the mouse button or press the Space Bar to display the answers.

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Objectives Solve quadratic equations by graphing Estimate solutions of quadratic equations by graphing

Vocabulary Quadratic equation – Roots – Zeros –

Addition and Subtraction PoE Properties of Equality (PoE) are based on the concept that as long as you do the same thing to both sides of an equation, then you have not changed anything. Addition PoE –For any numbers a, b, and c, if a = b, then a + c = b + c –You can add the same thing to both sides of an equation without changing it. Subtraction PoE –For any numbers a, b, and c, if a = b, then a - c = b - c –You can subtract the same thing from both sides of an equation without changing it.

Example 1 Solveby graphing. Graph the related function The equation of the axis of symmetry is or When f(x) equals orSo the coordinates of the vertex are

Example 1 cont Make a table of values to find other points to sketch the graph. xf(x) –38 –1–6 0–10 1–12 2 3–10 4–6 68 To solve you need to know where the value of f (x) is 0. This occurs at the x-intercepts. The x-intercepts of the parabola appear to be –2 and 5.

Example 1 cont CheckSolve by factoring. Original equation Factor. Solve for x. Zero Product Property or Answer: The solutions of the equation are –2 and 5.

Example 2 Solveby graphing. First rewrite the equation so one side is equal to zero. Original equation Add 9 to each side. Simplify. Graph the related function xf(x)

Example 2 cont Notice that the vertex of the parabola is the x-intercept. Thus, one solution is 3. What is the other solution? Try solving the equation by factoring. Original equation Factor. orZero Product Property Solve for x. There are two identical factors for the quadratic function, so there is only one root, called a double root. Answer: The solution is 3.

Example 3 Solveby graphing.Graph the related function Answer: The graph has no x-intercept. Thus, there are no real number solutions for the equation. xf(x) –36 –23 –

Example 4 xf(x) 02 1–1 2–2 3–1 42 Solveby graphing. If integral roots cannot be found, estimate the roots by stating the consecutive integers between which the roots lie. Graph the related function Notice that the value of the function changes from negative to positive between the x values of 0 and 1 and between 3 and 4.

Example 4 cont The x-intercepts of the graph are between 0 and 1 and between 3 and 4. Answer: One root is between 0 and 1, and the other root is between 3 and 4. xf(x) 02 1–1 2–2 3–1 42

Example 5 Model Rockets Shelly built a model rocket for her science project. The equation models the flight of the rocket, launched from ground level at a velocity of 250 feet per second, where y is the height of the rocket in feet after t seconds. For how many seconds was Shelly’s rocket in the air? You need to find the solution of the equation Use a graphing calculator to graph the related function The x-intercept is between 15 and 16 seconds.

Example 5 cont Answer:between 15 and 16 seconds

Summary & Homework Summary: –The roots of a quadratic equation are the x- intercepts of the related quadratic function Homework: –pg