Chapter 2 – Polynomial and Rational Functions 2.1 – Quadratic Functions

This function is the polynomial function The a’s are constants (normal numbers) The x’s are variables This polynomial is of degree n What do we call a polynomial of degree 1? Linear! What do we call a polynomial of degree 2? Quadratic!

Properties of a quadratic function: If a is positive, the parabola opens upward If a is negative, the parabola opens downward (0, c) is the y-intercept Ex: How does the graph of the function below relate to the graph of y = x2 ? y = -5(x+3)2 + 1 This function is shifted left 3 units, shifted up 1 unit, stretched by a factor of 5, and flipped vertically (opens downward)

Vertex form for a quadratic function: (h, k) is the vertex x = h is the line of symmetry To convert to standard form, complete the square! Ex: Write y = x2 – 4x + 5 in vertex form. Look at the first 2 terms and complete the square To do that, divide the b by 2 and square it! Since we added 4, we have to subtract 4 at the end… Then complete the square!

Ex: Write y = 2x2 + 4x – 8 in vertex form. Separate the first 2 term, then factor out a 2 from the first two terms… Divide the second term by 2 and square it… Since we added 2 1’s, we have to subtract 2 at the end… Then complete the square!

Ex: Write y + 5 = -3x2 + 15x in vertex form.

Write the equation in vertex form: Notice that the vertex is always at !

Write an equation in vertex form of a parabola with vertex (2,-3) passing through (5, 66).

If factoring doesn’t work, use quadratic formula! Ex: Find the x-intercepts of y = x2 – 4x – 5 . To find the x-intercepts, find where y = 0 Factor!!! When factored, we see that our x-intercepts are at x = -1 and x = 5. If factoring doesn’t work, use quadratic formula!