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# Quadratics

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• Include the line of symmetry Include x intercepts Include vertex Include y intercept
• If there is a c-value that will be the other x-intercept
• Remember its always positive even on the negative side.
• Cubics: down on negative, up on positive
• ### Quadratics

1. 1. Quadratic Functions <ul><li>By Chris Dudley and Jose Ortega </li></ul>
2. 2. What is a Quadratic Function? <ul><li>A quadratic function is any equation with a degree* of two. </li></ul><ul><li>In standard form a quadratic equation is ax²+bx+c. </li></ul><ul><li>A quadratic equation is a trinomial** expression because in standard form it adds three terms (ax², bx, and c). </li></ul>
3. 3. Quadratic Explosion Letter Part a Quadratic coefficient*** (what your are multiplying the value of x² by). x Indeterminate**** value (the value that changes along the x value of the graph). b The linear coefficient (what you are multiplying x by and adding to ax²). c The constant coefficient (what you are adding to the total value of ax²+bx).
4. 4. A Quadratic Graph <ul><li>When you graph a quadratic equation the line forms a symmetrical U-shape called a parabola (puh-rab-uh-luh). </li></ul><ul><li>Altering the A value causes the parabola ’ s curvature to change. </li></ul><ul><li>Altering the C value changes the height of the vertex. </li></ul><ul><li>Altering the B value changes the site of the vertex. </li></ul>
5. 5. Parts of a Graph <ul><li>The vertex is at the center point of the parabola where there is no other point on the parabola with the same Y-value. </li></ul><ul><li>The Y-intercept is the point where the line crosses the Y-axis at x=0 </li></ul><ul><li>The X-intercept is where the line intercepts the x-axis at Y=0. A parabola can have 1, 2, or no X-intercpets. If the vertex falls on the x-axis there is one X-intercept. If the vertex falls under the x-axis there are two X-intercepts. If the vertex is over the x-axis there are no X-intercepts (Note: these rules apply if the A-value is positive, if it is negative then vice-versa applies.) </li></ul>
6. 6. The A Value and Graphing <ul><li>The A value or quadratic coefficient is the value by which you multiply the solution of x² (remember order of operations, don ’ t multiply x by a and then square it). </li></ul><ul><li>The quadratic coefficient can never equal zero. If it were zero, it would make the product of ax² zero and leave only a linear expression (bx+c). </li></ul>
7. 7. The A Value and Curvature <ul><li>If the quadratic coefficient is over one then it makes the standard parabola (x²) steeper. This happens because it makes the solution of x² larger with multiplication. </li></ul><ul><li>If the quadratic coefficient is under one, then it makes the standard parabola less steep. This happens because whenever a fraction or decimal is multiplied it makes, that fraction of the solution to x². </li></ul>2x² .5x²
8. 8. Negative A Values <ul><li>A negative A value takes the standard parabola (x²) and flips it upside down. </li></ul><ul><li>This happens because any time a number is multiplied or squared by itself the rules of integers cause the value to be positive. However, when that positive value of x² is multiplied by a negative, the solution turns negative. </li></ul><ul><li>Negative A values can also change the curvature as well as flip a parabola. In this case values over -1 (these values do appear larger however because we ’ re in the negatives) but below zero cause the steepness to decrease. Values below -1 cause the steepness to increase. </li></ul>(-1)x²
9. 9. The B Value <ul><li>The B value or linear coefficient is the value that you multiply x by in the linear term (bx). You then add the value onto the value of ax². </li></ul><ul><li>Although it is hard to tell the exact way the linear coefficient will shape the graph, it is easy to find out the approximate shape of the graph will. </li></ul><ul><li>Unless there is a C-value, one of the x-intercepts will be zero. </li></ul>
10. 10. B Value Graphs <ul><li>A linear coefficient does not change the curvature of the parabola. It instead moves the whole parabola to the side and down some. </li></ul><ul><li>The quadratic coefficient will always move the vertex into the opposite integer on the x axis. </li></ul><ul><li>The reason why a parabola moves to a side and then down is because in the opposite integer of the linear coefficient the value will be negative. This causes a depression in the parabola until ax^2 can create a larger value. </li></ul><ul><li>With a negative value the b value moves the vertex into its own integer along the x-axis and then moves it up. </li></ul>Draw Graphs here:
11. 11. The C Value <ul><li>The c value or constant coefficient is what you finally add to the value of ax²+bx+c. </li></ul><ul><li>The constant coefficient only moves the vertex of the graph and all its points up and down. </li></ul><ul><li>Interesting fact: The C value can never subtract the absolute value***** of ax²+bx (-|ax²+bx|) because it would always end up creating a zero. </li></ul>
12. 12. The C Value Graph <ul><li>Adding a negative c value always moves a parabola down. Adding a positive always moves the parabola up. </li></ul><ul><li>The effects of the constant coefficient is not affected by any other values in the standard form. </li></ul>x²+(-1) Or x²-1 x²+1
13. 13. FAQs (Frequently Asked Questions) <ul><li>What ’ s the difference between a quadratic equation and an exponential one?: In an exponential equation the exponent changes. In a quadratic equation the growth factor****** changes. </li></ul><ul><li>What about equations with a degree of three?: Equations with a degree of three are completely different. They do not graph a parabola and do not have the same standard form. </li></ul><ul><li>What is the purpose of quadratic equations?: As we are learning, quadratic equations are often used in connection with the measurement of area of rectangles. Cubics are used to find the volume of a cube. </li></ul>
14. 14. Vocabulary <ul><li>*Degree: The highest power of any term of a polynomial (Algebra to Go). </li></ul><ul><li>**Trinomial: A polynomial (the sum of terms) with three terms (the product of constants and variables) (Algebra to Go). </li></ul><ul><li>***Coefficient: A numerical value in a term of an agebraic expression (Algebra to Go). </li></ul><ul><li>****Indeterminate: (Of an equation) able to be satisfied by more than one value (Dictionary.com) </li></ul><ul><li>*****Absolute Value (indicated with | |): The distance from zero on a number line. Absolute value is never negative (Algebra to Go). </li></ul><ul><li>******Growth Factor: The number that is multiplying by itself the amount of times the exponent states(my own definition). </li></ul>
15. 15. Sources <ul><li>Algebra to Go </li></ul><ul><li>Dictionary.com </li></ul><ul><li>Wolframalpha.com </li></ul><ul><li>Coolmath.com </li></ul><ul><li>Graphs from freemathhelp.com </li></ul><ul><li>Picture on slide one from Wikipedia.org </li></ul>