Int. Alg. NotesSection 8.4Page 1 of 7

Section 8.4: Graphing Quadratic Equations Using Transformations

Big Idea:The graph of any quadratic function is a parabola that can quickly be sketched by transforming the graph of the parabola y = x2with a vertical shift, a horizontal shift, a reflection about the x-axis, and/or a vertical stretch or compression.

Big Skill: You should be able to sketch a quadratic function by completing the square so that the function explicitly show the transformations to be made, and then using those transformations to make the sketch.

Definition: Quadratic Function

A quadratic function (in general form)is a function of the form where a, b, and c are real numbers and a 0. The domain consists of all real numbers. The point (0, c) is the y-intercept of the graph of the quadratic function. The graph of a quadratic function is always a parabola.

A quadratic function (in standard form) is a function of the form where a, h, and k are real numbers and a 0. The point (h, k) is the vertex of the graph of the quadratic function.

A quadratic function (in factored form) is a function of the form where a, is a real number (a 0), and r1 and r2 are complex numbers (remember that the real numbers are a subset of the complex numbers). The points (r1, 0) and (r2, 0) are the zeros of the graph of the quadratic function, provided r1 and r2 are real numbers.

Graph of the Quadratic Function

Here are some points on the curve y = x2:
x / y = x2
-3 / 9
-2 / 4
-1 / 1
0 / 0
1 / 1
2 / 4
3 / 9
Notice that the vertex is at (0, 0).
Notice that the graph “open up”.
Notice that the y-axis (eqn: x = 0) is an axis of symmetry. /

The graph of every quadratic function can be obtained by transforming the graph of y = x2with:

  1. a vertical shift,
  2. a horizontal shift,
  3. a reflection about the x-axis,
  4. and/or a vertical stretch or compression

Graph of the Quadratic Function (a vertical shift):

Adding a constant to the graph of y = x2 has the effect of shifting the graph vertically (up/down).
The graph of y = x2 + k is the graph of y = x2 shifted up by k units when k is a positive number.
The graph of y = x2 + k is the graph of y = x2 shifted down by k units when k is a negative number.
Notice that the vertex is at (0, k).
Notice that the graph still “opens up”.
Notice that the y-axis (eqn: x = 0) is still the axis of symmetry. /

Practice:

  1. Sketch a graph of y = x2 + 1.5 and y = x2 – 3.

Graph of the Quadratic Function (a horizontal shift):

Adding a constant to the x variable in y = x2 before squaring has the effect of shifting the graph horizontally (left/right).
The graph of y = (x – h)2 is the graph of y = x2 shifted right by h units when h is a positive number.
The graph of y = (x – h)2 is the graph of y = x2 shifted left by h units when h is a negative number.
Notice that the vertex is at (h, 0).
Notice that the graph still “opens up”.
Notice that the vertical line x = h is the new axis of symmetry. /

Practice:

  1. Sketch a graph of y = (x + 1.5)2 and y = (x – 3)2.

Graph of the Quadratic Function (a reflection about the x-axis, and/or a vertical stretch):

Multiplying the x variable by a constant in y = x2after squaring can reflectthe graph about the x-axis, or produce a vertical stretching of the graph.
The graph of y = ax2 is the graph of y = x2stretched vertically upwards when ais a positive number greater than 1.
The graph of y = ax2 is the graph of y = x2 compressed vertically upwards when a is a positive number less than 1.
The graph of y = ax2 is the graph of y = x2 reflected about the x-axis and stretched vertically downwards when a is a negative number less than -1.
The graph of y = ax2 is the graph of y = x2 reflected about the x-axis and compressed vertically downwards when a is a positive number greater than -1.
Notice that the vertex is at (0, 0).
Notice that the graph “opens up” when a>0 and “opens down” when a< 0.
Notice that the vertical line x = 0 is still the axis of symmetry. /

Practice:

  1. Sketch a graph of y = 1.5x2 and y = -¾x2.

Graph of the Quadratic Function in Standard Form:

 Locate the vertex of the graph at (h, k).
 If a > 0, sketch the parabola opening up, otherwise sketch it opening down.
 If |a| > 0, sketch the parabola vertically stretched, otherwise sketch it vertically compressed.
 The picture to the left shows the steps for graphing
:
  • Locate vertex at (3, -1)
  • Sketch parabola opening down
  • Sketch parabola vertically stretched.
/

Practice:

  1. Sketch a graph of .

Graph of the Quadratic Function in General Form:

 Convert the function to standard form
by completing the square
 Graph the function using transformations.
To graph :
/

Practice:

  1. Sketch a graph of .
  1. Sketch a graph of .
  1. Determine the equation of the graph shown below.

Algebra is:

the study of how to perform multi-step arithmetic calculations more efficiently,

and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.