Quadratic Functions

The following diagram shows how to use the vertex formula to convert a quadratic function from general form to vertex form. Scroll down the page for more examples and solutions for quadratic equations.

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Forms of Quadratic Functions
We can write quadratic functions in different ways or forms:

• General Form (Standard Form)
  
• Factored Form (Intercept Form)
  
• Vertex Form
  

1. General Form (Standard Form)
   ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0.
   For example, y = 2x² + 5x − 30
   Advantages:
   This is the most common form, making it easy to identify the coefficients a, b, and c.
   It’s the form used for applying the quadratic formula.
   It’s useful for quickly determining the y-intercept (when x=0, y=c).
   

2. Factored Form (Intercept Form):
   a(x - r₁)(x - r₂) = 0, where a ≠ 0, and r₁ and r₂ are the roots (x-intercepts) of the equation.
   For example, y = 2(x + 6)(x − 5).
   Advantages:
   It directly reveals the x-intercepts (which are also the roots when the function is zero) (r₁ and r₂). For example, the x-intercepts of y = 2(x + 6)(x − 5) are (−6, 0) and (5, 0)
   It’s helpful for quickly sketching the graph of the quadratic function.
   

3. Vertex Form:
   a(x - h)² + k = 0, where a ≠ 0, and (h, k) represents the vertex of the parabola.
   For example, y = 2(x + 6)² − 5.
   Advantages:
   It directly reveals the vertex of the parabola, (h, k), which is the maximum or minimum point of the function. If a is positive then it is a minimum vertex. It a is negative then it is a maximum vertex.
   It’s helpful for understanding the transformations of the basic quadratic function.
   

The following video looks at the various formats in which Quadratic Functions may be written as.

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General Form to Factored Form

The following videos show how to change quadratic functions from general form to factored form.

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General Form to Vertex Form by using the Vertex Formula

We can change a quadratic function from general form to vertex form by using the vertex formula.
Example of how to convert standard form to vertex form of a parabola equation.

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General Form to Vertex Form by Completing the Square

We can change a quadratic function from general form to vertex form by completing the square.

The following videos show how to use the method of Completing the Square to convert a quadratic function from standard form to vertex form.

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Key Features of Quadratic Equations

1. Vertex:
   The highest or lowest point on the parabola, depending on whether the parabola opens downward or upward.
   Found using the vertex form or the formula h = −b/(2a), k = f(h).
   
2. Axis of Symmetry:
   A vertical line that passes through the vertex, given by x = h x=h (from vertex form) or x = −b/(2a) (from standard form).
   
3. Roots (X-Intercepts):
   The points where the parabola crosses the x-axis.
   Found by solving f(x)=0.
   
4. Y-Intercept:
   The point where the parabola crosses the y-axis.
   Found by evaluating f(0).
   

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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