Standard Form

Standard Form is a way of expressing a quadratic function in the form:

• \(x\) represents the independent variable
• \(y\) represents the dependent variable
• \(a\) represents the stretch factor of the function and determines whether it opens upward or downward
• \(c\) represents the y-intercept

where \(a ≠ 0\) and all coefficients are real numbers.

Show Answer

First, we can create a table of values to identify where each of the main points are located:

X	-1	0	1	2	3	4	5
Y	8	3	0	-1	0	3	8

Next, we can create our graph based on these values:

Table of Values

Every quadratic equation is a relationship of \(x\) and \(y\) values. To create a table of values, we just have to pick a set of \(x\) values, substitute them into the equation and evaluate to get the y values. You could also read the points from a graph.

Enter the \(a\), \(b\), and \(c\) values below to create a table of values:

Converting Standard Form to Factored Form

Converting standard form to factored form is called factoring of quadratics.

1. Check if there are any common factors
2. Determine 2 integers that result in the sum of \(b\) and the product of \(ac\)
3. Factor the first and last pairs of terms separatately

Converting Standard Form to Vertex Form

The process of converting a quadratic function in Standard Form to Vertex Form is identical to Completing the Square.

1. Factor the \(a\) value (if any) out of the first 2 terms
2. Find the special value \((\cfrac{b}{2})^2\)
3. Add, the special number to the bracket
4. Subtract the expression by the special number by placing it outside the brackets and multiplying it by \(a\)
5. Simplify the constants outside the brackets
6. Factor the trinomial inside the brackets as the square of a binomial

Show Answer

First, we can create a table of values to better determine where each of the main points is located. As the time starts at \(0\), we won't need to account \(x < 0\):

X	0	0.5	1	1.5	2	2.5	3	3.5	4
Y	35	38.75	40	38.75	35	28.75	20	8.75	-5

We can now create our graph using our table of values:

Using our graph, we can identify the maximum height of the ball as \(40\;[m]\). It would take 1s for it to reach that height. Lastly, we can identify that the ball reaches ground level at \(3.5-4\;[s]\) (roughly \(3.8\;[s]\)).