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MTH 065 Elementary Algebra II

MTH 065 Elementary Algebra II. Chapter 11 Quadratic Functions and Equations Section 11.7 More About Graphing Quadratic Functions. Forms of Quadratic Functions. Standard form … f(x) = ax 2 + bx + c Graphing form … f(x) = a(x – h) 2 + k. How to convert between these forms ….

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MTH 065 Elementary Algebra II

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  1. MTH 065Elementary Algebra II Chapter 11 Quadratic Functions and Equations Section 11.7 More About Graphing Quadratic Functions

  2. Forms of Quadratic Functions • Standard form … f(x) = ax2 + bx + c • Graphing form … f(x) = a(x – h)2 + k How to convert between these forms …

  3. Graphing Form  Standard From • Square the binomial, remove grouping symbols, and combine like terms … f(x) = 2(x – 4)2 + 5 f(x) = 2(x2 – 8x + 16) + 5 f(x) = 2x2 – 16x + 32 + 5 f(x) = 2x2 – 16x + 37

  4. Standard Form  Graphing From • Factor a out of the terms w/ the variable, complete the square, factor, and add the constants … f(x) = 2x2 – 16x + 37 f(x) = 2(x2 – 8x) + 37 f(x) = 2(x2 – 8x + 42) + 37 – 32 f(x) = 2(x – 4)2 + 5

  5. Locating the Vertex from the Standard From f(x) = 2x2 – 16x + 37 f(x) = 2(x2 – 8x) + 37 f(x) = 2(x2 – 8x + 42) + 37 – 32 f(x) = 2(x – 4)2 + 5 • To get to the 4 (the x-coordinate of the vertex) … • Start with the opposite of the linear coefficient (b) … • Divide by the quadratic coefficient (a) … • Divide by 2 • i.e.

  6. Locating the Vertex from the Standard From f(x) = 2x2 – 16x + 37 f(x) = 2(x2 – 8x) + 37 f(x) = 2(x2 – 8x + 42) + 37 – 32 f(x) = 2(x – 4)2 + 5 • To get to the 5 (the y-coordinate of the vertex) … • Start with the constant term (c) … • Subtract a times the square of the x-coordinate … • i.e.

  7. Locating the Vertex f(x) = ax2 + bx + c Vertex:

  8. The y-intercept • Let x = 0 f(x) = ax2 + bx + c f(0) = a(0)2 + b0 + c f(0) = c • Therefore, the constant term is the y-intercept.

  9. The x-intercepts • Let f(x) = 0 f(x) = ax2 + bx + c 0 = ax2 + bx + c … solve for x (if possible) … • Therefore, there may be 0, 1, or 2 x-intercepts.

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