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Quadratic: Standard (Vertex) Form

y = a(x - h)2 + k

What to Learn

Help to Remember What to Learn

 

y = a(x - h)2 + k

  • If a < 0, the quadratic will open downwards.
  • If a > 0, the quadratic will open upwards.
  • Vertical Stretch:
    • Increasing the value of a will vertically expand the graph (ie, make the arms move closer together).
    • Decreasing the value of a will vertically compress the graph (ie, make the arms spread out wider).
    • When you move a to a negative value you will see that there is a reflection about the x-axis. Looking at the change in a relative to the x-axis will help you see the expansion/contraction better.
  • (h, k) will always be the coordinates of the vertex (the minimum/maximum value) of the quadratic function.
  • Domain: xR.
  • Range:
    • if a > 0, y > k
    • if a < 0, y < k
  • Note: When the form changes to a(x + h)2 + k, then (h, k) is the vertex, BUT, h is a negative number.
  • Note: When the form changes to a(x + h)2 - k, then (h, k) is still the vertex BUT k will have a negative value.

Quadratic Standard (Completed Square) Form Interactive Activity

Move the sliders for a, h, and k. Slider b will not be studied in Math 20 Pure.

  • Study the effect of changing a from positive to negative values.
  • Study the effect changing h and k has on the vertex (the minimum/maximum value) position.
  • Watch the operation for h in the (x - h) section of the top left corner of the data output.
  • Find the x-intercept(s) on the graph. These will be the solutions of the form: y = a(x - h)2 + k
  • Find the y-intercept on the graph. Notice that the a, b and k parameters do not help you determine the location of the y-intercept.

Linked Source - Ron Blond

Linked Source - Ron Blond

 

Question 1:

Find the equation of a quadratic function given the vertex, (2, -4) and the y-intercept of 4.

Sample Solution

y = a(x - h)2 + k

The question provides the vertex, (2, -4). Thus (h, k) = (2, -4).

y = a(x - 2)2 - 4

The y-intercept, (0, 4), is on the quadratic. Thus (x, y) = (0, 4). Substitute these values into the equation:

y = a(x - 2)2 - 4

4 = a [(0) - 2]2 - 4

a (-2)2 = 4+4

4a = 8

a = 2

Therefore, the equation of the quadratic is y = 2(x - 2)2 - 4

Note: The steps are similar if given the vertex and another point on the quadratic.

Question 2:

Write an equation for a parabola with vertex (3, -4) and passing through (2, -7).

Sample Solution

y = a(x - h)2 + k

The question provides the vertex, (3, -4). Thus (h, k) = (3, -4).

y = a(x - 3)2 - 4

(2, -7) is on the quadratic. Thus (x, y) = (2, -7). Substitute these values into the equation:

y = a(x - 3)2 - 4

-7 = a [(2) - 3]2 - 4

-7 = a (-1)2 - 4

a (-1)2 = -7 + 4

a = -3

Therefore, the equation of the quadratic is y = -3(x - 3)2 - 4

Note: The steps are similar if given the vertex and another point on the quadratic.

Enrichment: Quadratic General to Standard Form Proof

  • Domain: xR.
  • Range:
    • if a > 0, y > k
    • if a < 0, y < k

 

 
Comments to:  Jim Reed
Started September, 1998. Copyright 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007