Solving Quadratics


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Solving Quadratics

  1. 1. Quadratics! Three Forms and Graphing
  2. 2. Standard Form <ul><li>Standard Form is y=ax ²+bx+c where a does not equal 0 </li></ul><ul><li>Tricks for Factoring in this Form: </li></ul><ul><ul><li>If a and c are both perfect squares, try one of the special patterns: </li></ul></ul><ul><ul><ul><li>Difference of two squares: a²-b²=(a+b)(a-b) </li></ul></ul></ul><ul><ul><ul><ul><li>Ex. x²-4=(x+2)(x-2) Because the square root of x² is x and the square root of 4 is 2 </li></ul></ul></ul></ul><ul><ul><ul><li>Perfect Square Trinomial: Use if a=1 and b is a perfect square </li></ul></ul></ul><ul><ul><ul><ul><li>Ex. x²+6x+9=(x+3)² because the square root of x² is x, the square root of 9 is 3, and 3+3=6, which is the b term </li></ul></ul></ul></ul><ul><ul><li>Before factoring, check to see if the terms have a Greatest Common Factor </li></ul></ul>
  3. 3. Vertex Form <ul><li>Vertex Form: y=(x-h) ²+k </li></ul><ul><li>In order to factor from vertex form, it must be converted to standard form </li></ul>
  4. 4. Intercept Form <ul><li>Intercept Form: y=a(x-p)(x-q) </li></ul><ul><li>Just like vertex form, you must convert intercept form to vertex form in order to factor it </li></ul>
  5. 5. Graphing in Standard Form <ul><li>Tricks from the Equation: </li></ul><ul><ul><li>a will tell you if the graph opens up or down, which affects whether there is a minimum or a maximum </li></ul></ul><ul><ul><ul><li>If a is positive, the graph opens up, if a is negative, the graph opens down </li></ul></ul></ul><ul><ul><li>c will tell you the y-intercept of the graph </li></ul></ul><ul><li>Steps for Graphing: </li></ul><ul><ul><li>Find the vertex: (-b/2a, f(-b/2a)) </li></ul></ul><ul><ul><ul><li>To find the y-coordinate of the vertex, plug the x back into the equation and solve </li></ul></ul></ul><ul><ul><li>To find two more points, use info you already know: You know that c is where the graph crosses the y-axis, so use that as one point </li></ul></ul><ul><ul><li>To find the other point, you need to know the axis of symmetry. It’s simple to find the axis of symmetry; it’s simply a line running parallel to the y-axis that goes through the vertex. (You just need to know the x-coordinate of the vertex) </li></ul></ul><ul><ul><li>Once you have the line drawn, simply mark a point an equal distance from the line of symmetry that is equal to the distance from the y-intercept to the line of symmetry </li></ul></ul><ul><ul><li>Connect all of these points with a smooth line and label </li></ul></ul>
  6. 6. Graphing in Vertex Form <ul><li>Tricks from the Equation: </li></ul><ul><ul><li>a is has the same rules as a in standard form </li></ul></ul><ul><ul><li>The vertex is (h,k) </li></ul></ul><ul><ul><li>h determines how far left or right the parabola is </li></ul></ul><ul><ul><li>k determines how far up or down the parabola is </li></ul></ul><ul><li>Steps for Graphing: </li></ul><ul><ul><li>Determine and plot the vertex (h,k) </li></ul></ul><ul><ul><li>Draw the axis of symmety. The equation for this is x=h </li></ul></ul><ul><ul><li>Find coordinates of a point to the left and right of the vertex </li></ul></ul><ul><ul><ul><li>(A good trick is to go over left or right by one unit and up by the value of a (or go down if a is negative) however, this trick only works once) </li></ul></ul></ul><ul><ul><li>Connect the points and label </li></ul></ul>
  7. 7. Graphing in Intercept Form <ul><li>Tricks from the equation: </li></ul><ul><ul><li>p and q are the x-intercepts of the parabola </li></ul></ul><ul><ul><li>a is still the same as compared to standard and vertex form </li></ul></ul><ul><ul><li>The y-intercept is apq </li></ul></ul><ul><li>Steps for Graphing: </li></ul><ul><ul><li>Plot the x-intercepts </li></ul></ul><ul><ul><li>Find the coordinates of the vertex. Vertex=(p+q/2, f(p+q/2)) </li></ul></ul><ul><ul><li>Plot the vertex and the axis of symmetry (axis of symmetry equation is x=p+q/2) </li></ul></ul><ul><ul><li>Connect the points and label </li></ul></ul>
  8. 8. Now go watch the videos that demonstrate graphing!