Graphing a parabola from the vertex form

 

Basic form of parabola

• The basic form of parabola is y=x^2 (x to the power of 2).

• Therefore it starts at the origin (0,0) because the vertex is not given, meaning it is (0,0).

• Let's make a table for the y and x values...

 

 

 

 

 

 

 

 

 

 

 

    Question time- If the value of x=5 what would the value of y be?

     Answer is.... 25!!

 

The vertex form

• Written as 

 

 

• The "h" and "k" are the points for the vertex where "h" is the x-value and "k" is the y-value.

• For "h" the sign changes when finding the vertex. So, if it has a minus sign in front, it becomes plus sign, and if it is plus sign, it becomes minus sign. For example, if the equation was       y=a(x-3)^2+4, the "h" value (-3) would change to "3" when we plot the vertex on the graph.

• The "k" (y-value) stays the same. 

 

 

 

 

 

 

 

 

 

 

 

Question time- What would be the vertex for the equation y=a(x+2)^2-5?

 

Answer would be.... (-2,-5)!!

 

Now we know the vertex, but we need to find more points in order to graph the parabola.

How would we do that?

• Using the step pattern!!

• If the "a" value in the eqaution is positive and 1 then we follow the normal step pattern of "over 1, up 1, and over 2, up 4." However, remember that this pattern starts from the vertex.

• If the "a" value is more or less than 1 than we multiply that number by our normal step pattern.

 

Let's make a table to explain this, if the "a" value is 2.

 

 

 

 

 

 

 

 

 

 

 

 

This shows that if the "a" value is 2, then the step pattern is doubled (2 x normal step pattern).

 

Let's try one more example if the "a" value is 0.5.

 

 

 

 

 

 

 

 

 

 

 

 

This shows that if the value of "a" is 0.5, then the step pattern is halved (0.5 x normal step pattern).

 

Try this...

• Make a table for the "y-value" if "a" is equal to 3.

 

Answer is.....

 

 

 

 

 

 

 

 

 

 

 

 

Ready to graph!

• Once you have the vertex, and atleast two other points, you can graph the parabola. 

 

Time for examples!

Equation is y=2(x+3)^2-4

In this example, the x-value of the vertex is -3 (remember the rule of switching the sign), and the y-value is -4. 

• Therefore, vertex is (-3,-4).

• Now let's make a table for the step pattern!

 

 

 

 

 

 

 

 

 

 

 

 

 

The graph would look like...

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Another example?

Of course!

 

Equation is y=-0.5(x-2)^2+3

Tricky!

 

• If there is a negative sign in front of "a" it means that the graph is vertically reflected.

• Vertical reflection is when the parabola flips over vertically, and from the vertex the parabola moves down, instead of going up. 

• Therefore, if the sign in front of "a" is positive, the parabola moves upwards, and if it is negative, then it moves downwards.

 

Let's find the vertex!

The x-value would be 2 (remember the rule of switching the sign), and the y-value would be 3 (do not switch the sign of the y-value).

 

Vertex=(2,3)

Now let's make a table for the step pattern...

 

 

 

 

 

 

 

 

 

 

 

 

 

The graph would look like...

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Try these examples...

• y=1.5(x+5)^2-5

• y=-2(x+1)^2+5

• y=3(x-4)^2-2

• y=(x-2)^2-4

• y=-(x+5)^2+1

 

 

 

Try out this video for a better understanding!