MATH 100 Quadratic Function Worksheet

Name:
Names of people you worked with:
The purpose of this exercise is to understand the "standard form" of a quadratic function (p. 113) by relating it to the shifts and stretches discussed in section 1.6 of the textbook. Work in groups of two to fill in answers to the questions below. Each group member should contribute equally.
A polynomial function of the form f(x) = ax2 + bx + c is called a quadratic function. We already know that the x-intercepts of a quadratic function are given by the quadratic formula (or can be found by factoring). In this worksheet we learn that the graph of any quadratic function is a parabola (p. 55 (e)).

As defined on p. 111 of your text, the axis of symmetry of a parabola is the vertical line down its center; the parabola has mirror symmetry across the line of the axis of symmetry. The vertex is the point where the axis of symmetry intersects the parabola; it represents either a maximum or minimum value of the quadratic function whose graph is the parabola.

For instance, the quadratic function f(x) = x2 - 1 whose graph is shown on the left below has the axis of symmetry x=0 and vertex (0,-1).

The quadratic function g(x) = x^2 whose graph is shown on page 55 of your text has axis of symmetry x=0 and vertex (0,0). Since f(x) + 1 = g(x), this should be no surprise; adding 1 to the output of f(x) shifts the graph, and the vertex, upward by 1 unit. Similarly, if we define a function h(x) = f(x-2) = (x-2)2 - 1 = x2 - 4x + 3 will have the same graph as that of f(x) only shifted 2 units to the right; its axis of symmetry will be x=2 and its vertex will be at (2,-1). The graph of F(x) = 2(f(x)) = 2x2 - 2, shown on the right above, has the same axis of symmetry as that of f(x). However, multiplying the output of f(x) by 2 has "stretched" the graph vertically so that the vertex of F(x) is at (0,-2) and the parabola falls and rises more steeply.

Use reasoning similar to the examples given above and used in section 1.6 of your text to fill in the properties of the graphs of functions in the table below. Use a graphing calculator or Graphmatica to check your work.

Function Vertex Opens up/down Stretched?
f1(x) = x2      
f2(x) = x2 + 3      
f3(x) = x2 + k      
g1(x) = (x - 2)2      
g2(x) = (x - 2)2 + 3      
g3(x) = (x - h)2      
g4(x) = (x - h)2 + k      
g4(x) = (x - h)2 + k      
f4(x) = -x2      
f5(x) = 42      
f6(x) = 4(x - 2)2 + 3      
f7(x) = -4(x - 2)2 + 3      
f8(x) = a(x - h)2 + k      
       
As seen on page 113, f(x) = a(x - h)2 + k (where a is not 0) is the standard form for the equation of a quadratic function. All quadratic functions can be written in this form (you may prove this statement for extra credit!)

  1. What is the equation of the axis of symmetry of the graph of f(x) = a(x - h)2 + k?

     

  2. What are the coordinates of the vertex of the graph of f(x) = a(x - h)2 + k?

     

  3. The graph of f(x) = a(x - h)2 + k is a parabola; when will this parabola open up? When will it open downward?

     

     

  4. When will the graph of f(x) = a(x - h)2 + k have the same shape as graph (e) on page 55 and when will it be "stretched"?