Standard Form is a way of expressing a quadratic function in the form:
where \(a ≠ 0\) and all coefficients are real numbers.
First, we can create a table of values to identify where each of the main points are located:
Next, we can create our graph based on these 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 is called factoring of quadratics.
First, we can identify \(2\) as a common factor across all terms. We can factor out the \(2\) to simplify the function:
Next, we need to determine 2 integers that result in the sum of \(b\) (6) and product of \(ac\) (5). We can create a table of values to make this process easier for ourselves:
We can rewrite the function with these numbers in order to factor the 2 pairs of terms:
Therefore, \(2(x^2 + 6x + 5)\) converted to Factored Form is \(2((x + 1) + (x + 5))\).
The process of converting a quadratic function in Standard Form to Vertex Form is identical to Completing the Square.
As \(a = 3\), we can factor it out of the first 2 terms:
Next, as \(b = 3\), we can use this value to find the special number:
We can factor the trinomial inside the brackets by finding 2 numbers that result in the sum of the second term and the product of the third term:
Therefore, we can determine that function \(y = 3x^2 + 24x + 6\) converted to vertex form is \(y = 3(x + 4)²) - 42\).
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 \(x < 0\):
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 \(3.8\;[s]\)).