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Graphing Quadratic Equations

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What does a quadratic equation look like? One variable is squared No higher powers Standard Form y = ax 2 + bx + c y = x 2 – 4x + 1 What would the graph of this function look like?

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Graphing Quadratic Equations Standard form y = ax 2 + bx + c Example: y = x 2 – 4x + 1 x x 2 – 4x + 1 y -1 (-1) 2 – 4(-1) + 1 6 0 (0) 2 – 4(0) + 1 1 1 (1) 2 – 4(1) + 1 -2 2 (2) 2 – 4(2) + 1 -3 3 (3) 2 – 4(3) + 1 -2 4 (4) 2 – 4(4) + 1 1 Parabola Predict what y will be if x = 5?6

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Parabolas are commonly used in construction

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Many bridges use parabolas

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Parabolas can be fun

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Parabolas can be found in sports

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Parabolas occur in nature

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Parabolas can be used to solve many real world problems

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So, we are going to take the first step and study some of the basics A special plane is used to simulate weightlessness by flying in the arc of a parabola….

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Graphing Quadratic Equations Standard form y = ax 2 + bx + c Example: y = x 2 – 4x + 1 The graph of a quadratic Function is called a Parabola. It will open Upward when “a” is positive Graph this in Y1=

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Graphing Quadratic Equations Standard form y = ax 2 + bx + c Example: y = -x 2 – 4x + 1 Now graph the same Function, but make the “a” negative Graph this in Y2=

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y = -x 2 – 4x + 1 y = x 2 – 4x + 1 Parabolas highest or lowest point is a called the vertex. Maximum or a Minimum.

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y = -x 2 – 4x + 1 y = x 2 – 4x + 1 A vertical line that goes through the vertex is called the axis of symmetry. This can be found by using the formula

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describe the shape of the parabolas when the squared term was positive? __________________ describe the shape of the parabolas when the squared term was negative? __________________ 1.The solution to a __________________ function y = ax 2 + bx + c are the values of _______, when y = 0. 2. When graphing a quadratic function, the solutions are the ________ intercepts. At those points, the y coordinates are equal to ________. This explains why the x intercepts are also called the ______________. 3. The solutions to quadratic functions are also sometimes called _____________, the origin of something. Open upward Smile Open downward Frown Quadratic x x 0 Zeros roots

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4a. # of real solutions ______ 4b. # of real solutions _______ 4c. # of real solutions _______ 5a. solution: {__________} 5b. solution: {__________} 5c. solution: {__________} 102 3 -2, 1

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System of Equations Systems of equations 2 or more equations on the same graph Solution to a system the Point or points that the equations have in common Solve by Graphing Graph the equations and find the intersection point(s)

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Solving a system of equations First clear out Calculator 2 nd + 7 1 2

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1. Enter equations in Y= 2. Press Graph 3. Select intersect, press enter 4. Press Enter 5. Press Enter 6. Use arrow keys to get Close and press enter 7. Intersection point shown on bottom 8.Repeat steps 3-6.Use arrow keys to move up, Press Enter 9.2 nd intersection point shown on bottom

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Solving a system by substitution This one is a quadratic, graph will be a parabola This one is linear, they may intersect in 0, 1 or 2 pts Note they are both equal to y. To solve this, I can substitute the 2x – 4 for the y in the Other equation because 2x – 4 does equal y. It says so…. So when I rewrite the top equation it looks like this It’s a quadratic, to solve it, find out what x is, I have to Make it equal to zero. Now I can use whatever method I want to solve it, It looks like it would be easy to factor

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Now I can use whatever method I want to solve it, It looks like it would be easy to factor Use the zero product property to find each value of x Substitute each x into the linear equation (that’s the Easier one usually)to find the Y that goes with it. The parabola and the line intersect at 2 points (-7, -18) (8, 12) Graph and check it.

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