quadraticfunctions – Manroop's Blog

This week in Pre-Calculus 11 we began the Analyzing Quadratic Functions unit. This week we learned about the different properties of quadratic functions. To graph a quadratic function it is very important to have the ability to identify the properties of a quadratic function by looking at the equation.

What is General Form? General form is $ax^2+bx+c$ where c represents the y-intercept. In general form “a” represents the direction the parabola will open (if “a” is positive the parabola will open up and if “a” is negative the parabola will open down). “A” also represents the scale of the parabola and whether the parabola will be compressed or stretched.

The Scale of a Parabola: The scale of a parabola can be determined by the coefficient in front of the $x^2$ . If the coefficient is 1 the scale of the parabola will be 1-3-5 which means it is congruent to the parent function ( $y=x^2$ ). The 1-3-5 pattern means that when you start at the vertex you go one to the right and rise one and then you go one to the right and rise 3… If the coefficient of $x^2$ is not 1 then you just have to multiply 1-3-5 by the coefficient to figure out the scale.

Vertex: The vertex is the highest or lowest point of the parabola.

Axis of Symmetry: The axis of symmetry intersects the parabola at the vertex. The axis of symmetry splits the parabola exactly in half.

Minimum and Maximum Point: The minimum and maximum point of the parabola is also determined by the coefficient on $x^2$ . The minimum and maximum point is whether the vertex of the parabola is at the highest or lowest point it could possibly be at. If the vertex is at it’s minimum point that means it opens up and the coefficient on $x^2$ is positive. If the vertex is at it’s maximum point that means it opens down and the coefficient on $x^2$ is negative.

Domain: The domain of a quadratic function is all the possible values of x, the domain is alway $x\in\Re$ .

Range: The range is all the possible y values, this is dependent on the vertex.

Ex. $y=x+4x-3$

y intercept= -3

Congruent to $y=x^2$

Scale: 1-3-5

Minimum Point (opens up)

Domain: $x\in\Re$

Ex. $y=-7x^2+5x-8$

y-intercept= -8

Not congruent to $y=x^2$

Scale: -7- -21 – -35

Maximum Point (opens down)

Domain: $x\in\Re$

What is Standard Form? Standard Form is $y=a(x-p)^2+q$ . Standard form is also known as vertex form because p and q represent the vertex’s coordinates. P also tells us the horizontal translation and q also represents the vertical translation.