Week 9 – Pre Calc 11 – Manroop's Blog

This week in Pre-Calculus 11 we finished the Analyzing Quadratic Functions unit. This week we learned about factored form and modelling problems with quadratic functions. Factored form is very helpful when looking for the x-intercept and it can also help with finding the axis of symmetry.

What is Factored Form? The formula for factored form is $y=a(x-x_1)(x-x_2)$ . This formula is very useful when looking for the roots of a quadratic function. You can easily convert from general form to factored form by factoring and finding the solutions of the function.

Ex. $y=x^2+6x-16$

-1 x 16= -16 -1+16= 15

-2 x 8= -16 -2+8= 6

-4 x 4= -16 -4+4= 0

$y=(x+8)(x-2)$

$x+8=0$

x= -8

$x-2=0$

x= 2

x-intercepts= (-8,0) and (2,0)

y-intercept= (0, -16)

Opens up (minimum)

Congruent to $y=x^2$

How to Find the Axis Of Symmetry: To find the axis of symmetry you have to find the average of the x-intercepts by adding them together and then dividing the sum by two. After finding the axis of symmetry you can find the vertex by plugging the x value into the equation.

Ex. $y=2x^2+7x-4$

$y=(2x-1)(x+4)$

$2x-1=0$

$2x=1$

$x=\frac{1}{2}$

$x+4=0$

$x=-4$

x-intercepts= $(-4,0)$ and $(\frac{1}{2},0)$

AOS: $x=\frac{\frac{1}{2}+-4}{2}$

AOS: $x=\frac{\frac{1}{2}+\frac{-8}{2}}{2}$

AOS: $x=\frac{\frac{-7}{2}}{2}$

AOS: $x=\frac{-7}{4}$

$y=(2(\frac{-7}{4}-1)(\frac{-7}{4}+4)$

$y=(\frac{-7}{2}-\frac{2}{2})(\frac{-7}{4}+\frac{16}{4})$

$y=(\frac{-9}{2})(\frac{9}{4}$

$y=\frac{-81}{8}$

Vertex: $(\frac{-7}{4},\frac{-81}{8})$

How to Deal with Word Problems: To deal with word problems involving quadratic functions the first step is to find a relationship between the variables. After finding the relationship between the variables find the x-intercept. Then find the axis of symmetry and plug it into the equation to find the y value. The answer of the word problem is going to involve the vertex of the quadratic function.

Ex. Find two integers with the difference of 11 and the greatest product.

$x-y=11$