The Standard forms of a Parabola and an Inquiry into Their Interrelatedness

by: Al Byrnes

Show and describe the relationship between these two forms of the graph of a parabola:

y = ax² + bx + c (standard form)

y = a(x - h)² + k (vertex form)

Show the derivation of one equasion from the other and give an interpretation for the value of the parameters *a, b, c, h, *and *k.*

Part 1: show the derivation of the standard form of a parabola from the vertex form for a parabola:

y = a(x - h)² + k

= a((x - h)(x - h)) + k

= a(x² - hx -hx + h²) + k

= a(x² - 2hx + h²) + k

= ax² - 2ahx + ah² + k

= ax² + (-2ah)x + (ah² + k)

Let b = (-2ah) and c = (ah² + k)

So after substitution: y = a(x - h)² + k = ax² + bx + c

What is the relationship between the different parameters in the two standard forms? Lets begin the discussion with a reminder of the meaning of the parameter values *h *and *k*, from the vertex form of the parabola. *h *and *k *are the x and y coordinates of the parabola's vertex, respectively. The graph below provides an illustration of the parabola y = (x + 1)² + 1 (vertex form)

Note that if we consider the general vertex form of the parabola y = a(x - h)² + k with the parabola sketch above, we can confirm that the vertex is at (*h, k*) or (-1, 1) as expected. Also note that in our example, the value of the parameter *a *is 1.

Using the relationships established above, let's try to determine coeffiecents* c *and *b* to be used in the standard form of the equasion for the same parabola:

*b *= (-2ah); *c *= (ah² + k)

To determine the value of parameter *c *for this example, let's plug in the values for the parameters *a, h, *and *k *from our example, where *a = *1, *h = *-1, and *k *= 1:

*c = *(ah² + k); *c *= (1•(-1²) + 1)

*c *= (1•1 + 1)

*c *= (1 + 1)

*c *= 2

The number two is also significant when considering the graph of our parabola y = x² + 2x + 2 or y = (x + 1)² + 1; how is the value 2 pertainate to the graph below?

The value of the parameter *c *appears to correspond with the value of the y-intercept of the parabola's graph. Said another way, a parabola will intersect with the y-axis at *c *= (ah² + k). This is pretty cool as we can see from this equasion how the leading coefficient *a, *and the x and y coodinates of the vertex (*h *and *k*, respectively) are multiplied and added to determine the y - intercept of the parabola.

Nice. What about the parameter *b *from the standard form of the parabola? Recall that we determined by algebraically manipulating the vertex form of the parabola that the value of parameter *b *could be determined as a product of the leading coefficient, the x coordinate of the parabola's vertex and -2. So let's try it out. Given our vertex form and the parameter values extracted from that vertex form, here are the necessary steps to finding the value for parameter *b*:

vertex form of parabola: y = (x + 1)² + 1

*b = *(-2ah); a = 1, h = -1

*b *= (-2•1•(-1))

*b *= 2

Here we can see how the parameters *a, h, *and -2 influence the value of the parameter *b *in the standard form of a parabola.

We have additionally ascertained the values of *a, b, *and *c *necessary for representing the parabola expressed in vertex form in standard form:

y = x² + 2x + 2

Write up 2: part 2. Give an interpretation for the value of the parameters *a, b, c, h, *and *k.*

*To be continued...*

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