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- Watch and Read
- - Let's Start Thinking
  - Describing Transformations
  - Graphing Given Vertex Form
  - Finding an Equation From a Graph
  - Wrap-Up
  - Alternative Format
- Review
- - Check Your Understanding 1
  - Check Your Understanding 2
  - Explore This 1
  - Check Your Understanding 3
  - Check Your Understanding 4
  - Check Your Understanding 5
  - Check Your Understanding 6
  - Check Your Understanding 7
- Practise
- - Exercises
  - Answers
  - Solutions

Graphing Given Vertex Form

Explore This 1

Created with GeoGebra. Author: University of Waterloo. CC BY-NC-SA 4.0. https://ggbm.at/svfs9wee#

Explore This Summary

You may have noticed that:

All three graphs have the same "shape."
The horizontal and vertical distances between the marked points and the vertex were the same for all three graphs.
Changing the \(a\)-value changed the vertical distances between the marked points and the vertex.
The larger vertical distance displayed on the graph was always four times the smaller distance.

How the \(a\)-Value Affects the Shape of the Graph

Use this table to relate the shape of the graph of a quadratic relation to the value of \(a\) in its equation:

If \(a \gt 0\), these points are above the vertex.
If \(a \lt 0\), these points are below the vertex.

Horizontal Distance from the Vertex	Vertical Distance from the Vertex
\(1\)	\(a\)
\(2\)	\(4a\)
\(3\)	\(9a\)

\(a \gt 0\)

The value of a is greater than 0 so the points are above the vertex.

\(a \lt 0\)

The value of a is less than 0 so the points are below the vertex.

Notice that the vertical distance is always \(a\), multiplied by the square of the horizontal distance.

Graph Sketching

Two ways to graph a quadratic relation from its equation in vertex form:

Start with the graph of \(y=x^2\).
Apply transformations to create the graph of \(y=a(x-h)^2+k\).

Start at the vertex.
Use the \(a\)-value to determine where other key points on the parabola lie.

Plotting Plots in Relation to the Vertex

\(a \gt 0\):

\(a \lt0\):

Example 3 — Part A

Sketch the graph of each relation using at least five points.

\(y=(x-5)^2-3\)

Solution — Part A

Horizontal Distance from the Vertex	Vertical Distance from the Vertex
\(1\)	\(a=1\)
\(2\)	\(4a=4\)

Example 3 — Part A Continued

Sketch the graph of each relation using at least five points.

\(y=(x-5)^2-3\)

Example 3 — Part B

Sketch the graph of each relation using at least five points.

\(y=-(x+3)^2\)

Solution — Part B

Horizontal Distance from the Vertex	Vertical Distance from the Vertex
\(1\)	\(-1\)
\(2\)	\(4(-1)=-4\)

Example 3 — Part C

Sketch the graph of each relation using at least five points.

\(y=4x^2-9\)

Solution — Part C

Horizontal Distance from the Vertex	Vertical Distance from the Vertex
\(1\)	\(4\)
\(2\)	\(4(4)=16\)

Example 3 — Part D

Sketch the graph of each relation using at least five points.

\(y=-2(x-4)^2+6\)

Solution — Part D

Horizontal Distance from the Vertex	Vertical Distance from the Vertex
\(1\)	\(-2\)
\(2\)	\(4(-2)=-8\)

Example 3 — Part E

Sketch the graph of each relation using at least five points.

\(y=\dfrac14(x+1)^2+5\)

Solution — Part E

Horizontal Distance from the Vertex	Vertical Distance from the Vertex
\(1\)	\(\dfrac14\)
\(2\)	\(4 \left(\dfrac14 \right)=1\)

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Check Your Understanding 3

In this exercise, you will practise graphing quadratic relations with no vertical stretch or compression.

Created with GeoGebra. Author: University of Waterloo. CC BY-NC-SA 4.0. https://ggbm.at/qe79zxka#

Check Your Understanding 4

In this exercise, you will practise graphing quadratic relations that include a vertical stretch.

Created with GeoGebra. Author: University of Waterloo. CC BY-NC-SA 4.0. https://ggbm.at/sxwrd5zh#

Check Your Understanding 5

In this exercise, you will practise graphing quadratic relations that include a vertical compression.

Created with GeoGebra. Author: University of Waterloo. CC BY-NC-SA 4.0. https://ggbm.at/b9ny6qup#

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