Exploring Parabolas

By Annie Sun

For a given point, called the focus, and a given line not through the focus, called the directrix, a parabola is the locus of points such that the distance to the focus equals the distance to the directrix.  A parabola is a U-shaped curve with those specific properties and can be defined with a quadratic equation.

So, let’s explore some parabolas and see if we can determine their qualities based on their graphs and corresponding equations.

We’ll start with the standard form of a quadratic equation:

Where a, b, and c are rational numbers and a cannot equal 0.

Let’s take a look at a simpler quadratic equation:

If a equals 1, we get a graph that looks like this:

If a equals -1, we get a graph that looks like this:

So, positive values of a have an upwards shape and negative values “flip” the U downwards.

Now let’s see what happens when we change the values for a

As a values get larger, the graphs become “skinnier.”  This makes sense because the x value is being multiplied by a, thus larger values for the y value.

Conversely, as a gets smaller, the graphs become “wider,” because the x values are being multiplied by fractions and the y values become smaller.

So the a value in the equation affects the horizontal stretch. Now that we understand what happens for values of a in a simple quadratic equation of

Let's take a look at

where a and b are rational numbers

The b values seem to affect the horizontal and vertical shift of the graph, positive values move the parabola to the left and negative values move the parabola to the right.

Then when we add in the c value of the equation in the form

The value of c in the equation is affecting the vertical shift of the parabola, positive values of c are translating the parabola upward and negative values downwards.

Let’s look at the standard form:

We know that the graph will look like a parabola and the a value will determine if it is going upward/downward and how skinny/wide it is.

If the parabola is going upward then that means there will be a minimum point and if the parabola is going downward, then there is a maximum point.  This point is called the vertex.

Let’s call this vertex point (h, k).  We also know that the axis of symmetry for the parabola will by the line where x = h.

Now if we were to solve the equation
This could be done by factoring, completing the square, or using the quadratic formula.  I will not describe each of the steps here, as it is just an Algebra lesson, but we will use the fact that after solving the quadratic equation we get the x value from

Then we can get the y value by plugging in what we got for x into the quadratic, shown next.

After the Algebra part, the quadratic can be written as a function of x:

Where h =and k = f(h)

So now let’s make some graphs!

Example 1:

So a = -2, b = 4, and c = 1

We know that since a = -2, the parabola will open downward and it will be “skinnier,” the b and c values will shift the graph.  Next let’s solve for h:
h = -b/2a = -4/2(-2) = -4/-4 = 1
and if h = 1, then k = f(1) = -2(1)² + 4(1) + 1 = -2 + 4 + 1 = 3

So the vertex for the parabola is at (1, 3) and the axis of symmetry is the line x = 1.

Example 2:

So, a = 1/2, b =-1, and c = -3

From what we know about positive and fractional values of a, we can picture this graph opening upwards and having a larger horizontal stretch, making it 'wider," the b value should shift the parabola to the right, and the c value will shift it downward.

Next, we can use the (h, k) formulas to give us an idea of where the vertex will be, which we know will be the minimum point, because the parabola is opening upwards.

h = -b/2a = -(-1)/1(1/2) = 1

f(1) = 1/2(1)^2 - 1 - 3 = -3.5

So our vertex is at (1, -3.5)

Let's look at the graph:

Example 3:

What do you think the graph of the equation below will look like?

How will the a value affect the parabola?

What will the b value do to the graph? The c value?