and the properties of their graphs such as vertex and x and y intercepts are explored, interactively, using an applet.
The graph of a quadratic function is "U" shaped and is called a parabola. The exploration is carried by changing the values of all 3 coefficients $a$, $h$ and $k$. Once you finish the present tutorial, you may want to go through another tutorial on graphing quadratic functions. This form of the quadratic function is also called the vertex form.
A - Vertex, maximum and minimum values of a quadratic function
The term (x - h) (x - h) If you multiply both sides of the above inequality by coefficient case 1: a is positive Add k to the left and right sides of the inequality a(x - h) The left side represents f(x), hence f(x) ≥ k. This means that k is the case 2: a is negative Add k to the left and right sides of the inequality a(x - h) The left side represents f(x), hence f(x) ≤ k. This means that k is the
a(x - h)
a(x - h)
Note also that k = f(h), hence point (h,k) represents a
minimum point when a is positive and amaximum point when a is negative. This point is called the vertex of the graph of f.Example: Find the vertex of the graph of each function and identify it as a minimum or maximum point. a) f(x) = -(x + 2) b) f(x) = -x c) f(x) = 2(x - 3)
a) f(x) = -(x + 2)2 - 1
b) f(x) = -x2 + 2
c) f(x) = 2(x - 3)2
a = -1 , h = -2 and k = -1. The vertex is at (-2,-1) and it is a maximum point since a is negative.
a = -1 , h = 0 and k = 2. The vertex is at (0,2) and it is a maximum point since a is negative.
a = 2 , h = 3 and k = 0. The vertex is at (3,0) and it is a minimum point since a is positive.
Use the html 5 (better viewed using chrome, firefox, IE 9 or above) applet below to explore the graph of a quadratic function in vertex form: f(x)=a (x-h)
1 - Use the boxes on the left panel of the applet to set a to -1, h to -2 and k to 1. Check theposition of the 2 - Set a to -1, h to 0 and k to 2. Check the position of the 3 - Set a to 2, h to 3 and k to 0. Check the position of the 4 - Set h and k to some values and a to positive values only. Check that the 5 - Set h and k to some values and a to negative values only. Check that the
B - x intercepts of the graph of a quadratic function in standard form
The x intercepts of the graph of a quadratic function f given by
are the
solutions, if they exist, of the quadratic equation
add -k to both sides a(x - h) (x - h) The above equation has real solutions if - k / a is positive or zero. The solutions are given by x
x2 = h - √(- k / a)
Find the x intercepts for the graph of each function given below a) f(x) = -2(x - 3) a) To find the x -2(x - 3) -2(x - 3) (x - 3) two real solutions: x The graph of function in part a) has two x b) We solve -(x + 2) one repeated real solution x The graph of function in part b) has one $x$ c) We solve 4(x - 1) - k / a = - 5 / 4 is negative. The above equation has no real solutions and the graph of function h hasno x intercept.
b) g(x) = -(x + 2)2
c) h(x) = 4(x - 1)2 + 5
1 - Go back to the applet window and set the values of a, h and k for each of the examples inparts a, b and c above and check the the x 2 - Use the applet window to find any x 3 - Use the applet window and set a and k to values such that -k / a < 0.How many 4 - Use the applet window and set k to zero.How many 5 - Use the applet window and set a and k to values such that - k / a > 0.How many
a) f(x) = 5(x - 3)2 + 3
b) g(x) = -(x + 2)2 + 1
c) h(x) = 3(x - 1)2
C - From vertex form to general form with a, b and c.
Rewriting the vertex form of a quadratic function into the general form is carried out by expanding the square in the vertex form and grouping like terms.
Rewrite f(x) = -(x - 2)2 - 4 into general form with coefficients a, b and c. Expand the square in f(x) and group like terms f(x) = -(x - 2) = - x A tutorial on how to More references on quadratic functions and their properties.
- Find Vertex and Intercepts of Quadratic Functions - Calculator: An applet to solve calculate the vertex and x and y intercepts of the graph of a quadratic function.
- Match Quadratic Functions to Graphs. Excellent activity where quadratic functions are matched to graphs.
- graphing quadratic functions .
- quadratic functions (general form) .
