Topic 2 Summary On Quadratic Functions

Topic 2 Summary On Quadratic Functions

Solving Quadratic Equations

Writing a quadratic function in vertex form

Write a quadratic equation in vertex form for the parabola that has a vertex at (-2, 3) and goes through the point (2, 15)

Multiple Choice

What is the equation of the quadratic function that has a vertex of (-4, 6) and goes through the point (-8, 26)

•  $f\left(x\right)\ =\ \frac{5}{4}\left(x-4\right)^2+6$  

•  $f\left(x\right)=\frac{5}{4}\left(x+4\right)^2+6$  

•  $f\left(x\right)=-\frac{5}{4}\left(x+4\right)^2+6$  

•  $f\left(x\right)=\frac{5}{4}\left(x+4\right)^2-4$  

Multiple Choice

What is the equation of the quadratic function that has a vertex of (3, -2) and goes through the point (5, 10)

• $f\left(x\right)=-3\left(x+3\right)^2-2$

• $f\left(x\right)=3\left(x-3\right)^2-2$

• $f\left(x\right)=3\left(x+3\right)^2+2$

• $f\left(x\right)=3\left(x-3\right)^2+2$

Completing the Square

Completing the square requires us to find a value for c that makes a perfect square trinomial in the form

 $f\left(x\right)=a\left(x^2+bx+\left(\frac{b}{2}\right)^2\right)+c-a\left(\frac{b}{2}\right)^2$  

 $f\left(x\right)\ =\ a\left(x+\frac{b}{2}\right)^2+c-a\left(\frac{b}{2}\right)^2$  

Completing the Square

 $f\left(x\right)\ =\ 2x^2+8x+6$    Given
 $f\left(x\right)=2\left(x^2+4x\right)+6$  get x² and x by themselves
 $f\left(x\right)=2\left(x^2+4x+\left(\frac{4}{2}\right)^2\right)+6-8$  balance equation
 $f\left(x\right)=2\left(x+2\right)^2-2$  rewrite as perfect square binomial

Multiple Choice

Complete the square of the following quadratic function: $x^2-4x+5=0$  

•  $2-i$  

•  $4-i$  

•  $2+i$  

•  $4+i$  

Multiple Choice

Complete the square of the following trinomial: $f\left(x\right)=3x^2+12x-6$  

•  $f\left(x\right)=3\left(x+2\right)^2-18$  

•  $f\left(x\right)=3\left(x+4\right)^2-18$  

•  $f\left(x\right)=3\left(x-2\right)^2-18$  

•  $f\left(x\right)=3\left(x-2\right)^2-6$  

Completing the Square with imaginary numbers

When a parabola does not cross the x-axis, we will get a complex number when we complete the square.

Completing the square with a complex number

 $\ x^2+6x\ +12=0$  
 $x^2+6x=-12$  
 $x^2+6x+\left(\frac{6}{2}\right)^2=-12+9$  
 $\left(x+3\right)^2=-3$  
 $x+3=\pm\sqrt{-3}$  
 $x=-3\pm i\sqrt{3}$  

Factoring Complex Binomials

 $9x^2+36$  
 $\left(3x-6i\right)\left(3x+6i\right)$  
 $9x^2+18i-18i-\left(6i\right)^2$  
 $9x^2-\left(-36\right)$  
 $9x^2+36$  

Fill in the Blanks

Factor  $25x^2+49$  

Finding the Intersection of a Linear and Quadratic Equation

How do you find the solution to a system of two equations, one being linear and the other a quadratic?

Finding the Intersection of a Linear and Quadratic Equation

Find the solution to the following equations

 $f\left(x\right)\ =\ x^2+12$  and  $g\left(x\right)\ =\ 4x\ +17$  
 $x^2+12=4x+17$  set each equation equal to each other
 $x^2-4x-5=0$  combine all terms to one side of the equation
 $\left(x-5\right)\left(x+1\right)=0$  Find the value(s) of x that make equation true
 $y=4\left(5\right)\ +17\ or\ y\ =\ 4\left(-1\right)+17$  substitute into one equation
 $y\ =\ 37\ or\ y\ =\ 13$  Solve for y