4.12 Chapter Summary
In this chapter, we studied quadratic functions both algebraically and graphically.
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| Introduction | In this chapter we'll explore quadratic relationships, relationships that look like either the letter "u" or like an upside-down "u" when graphed. | ||||||||
| The Square Function | The square function, f(x)=x^2 is the most basic type of "squared" relationship. The graph of this function looks like:
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| Adding Constants to the Square Function | If a >0 then f(x)=x^2+a is a square function shifted up "a" units. If a <0 then f(x)=x^2+a is a square function shifted down "a" units. |
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| Square Function Times a Constant | If a >0 then f(x)=ax^2 is a square function that looks like a "u". If a <0 then f(x)=ax^2 is a square function that looks like an upside-down "u". |
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| Quadratic Functions | The formula: |
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| Basic Factoring | The only way for a product to equal 0 would be if at least one of its factors is 0. Therefore, if we can rewrite an equation as a product, such as |
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| Multiplying Out Binomials: Using Geometry | We can multiply out binomials by thinking of them as the lengths of the sides of a rectangle. | ||||||||
| The FOIL Method | Using the FOIL method: "First, Inner, Outer, Last", we can multiply out any two binomials. The process is developed using the distributive property of numbers. | ||||||||
| Factoring Full Quadratics | In trying to factor a quadratic x^2+bx+c, we look for two numbers that multiply together to give us "c" and add together to give us "b". | ||||||||
| Solving "X Squared = Number" | The solution to the equation where a≥0 is x=\pm\sqrt{a} |
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| Completing the Square an Introduction | The method of completing the square can be used to turn any quadratic into (something)2=number. Once in this form, we can just take the square root of both sides to solve for our unknown variable. To complete the square for a quadratic x^2+bx+c take half of the "b" term, square it, then add and subtract this quantity to the quadratic. Note that the leading coefficient must be 1 for this method to work! |
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| Completing the Square Example | The method of completing the square can be used to solve any quadratic equation. | ||||||||
| The Quadratic Formula | The quadratic formula, can be used to solve any equation that looks like |
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| The Vertex Formula | The x-coordinate of the vertex of a parabola is found by the formula To find the y-coordinate of the vertex, we plug the x-coordinate back into the original function. |
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| Generalizing Quadratics | In addition to using quadratics to model physical phenomena, quadratics can be studied as mathematical creatures in their own right. In doing so, we can study the discriminant of the quadratic formula.
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