Standard Form

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

\(y = ax^2 + bx + c\)

  • \(x\) 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.

Sketch the quadratic function \(x^2 - 4x + 3\)

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

Convert \(2x^2 + 12x + 10\) from Standard Form to Factored Form.

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

Convert \(y = 3x^2 + 24x + 6\) from Standard Form to Vertex Form.

A ball is thrown upward with an initial velocity of \(10\;[m/s]\). Its approximate height \(h\), in metres, above the ground after \(t\) seconds is given by the relation \(h = -5t^2 + 10t + 35\).
1. Sketch a graph of the quadratic relation.
2. Find the maximum height of the ball.
3. Find how long it takes the ball to reach the maximum height.
4. Find when the ball is at ground level.

Try these questions: