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ACCELERATED PRE-CALCULUS SUMMER STUDY GUIDE

CHAPTER 23: QUADRATICS AND EQUATIONS IN THE FORM OF y = ax2 + bx + c

 

 

SECTION 23.1 WHAT IS A QUADRATIC SET EQUAL TO Y?

WHAT IS A QUADRATIC EXRESSION?

 

An algebraic expression in the form of ax2 + bx + c, where a, b and c are real numbers and a ≠ 0, is called a "quadratic expression."

Please note that all the terms in the quadratic expression are ADDED. Those terms must be added in order to accurately isolate a, b and c.

Here are some examples:

ex. 1) 2x2 + 2x + 5   
a = 2, b = 2 and c = 5

ex. 2) 3x2 - 3x + 4
I will rewrite 3x2 - 3x + 4 as 3x2 + - 3x + 4
a = 3, b = -3 and c = 4

ex. 3) -3x2
I will rewrite 3x2 as 3x2 + 0x + 0
a = -3, b = 0 and c = 0

 

 

Equations written in the form of y = ax2 + bx + c, where a, b and c are real numbers and a ≠ 0 are called "quadratics set equal to y." This type of "quadratic equation" will create ordered pairs that produce a figure called a "parabola."

AN ASIDE: YOU HAVE BEEN FACTORING QUADRATIC EXPRESSIONS ON MANY OCCASIONS!


Figure


Figure

HOW TO FIND THE VERTEX OF A PARABOLA

 

Xvertex= -b/(2a)    

Yvertex: substitute the Xvertex into the x values of the equation of the parabola.

 

Here are some examples of the above ideas:

ex. 1) Name a, b, and c, the vertex, and 4 ordered pairs of the parabola y = x2 + 1x + 1 and then graph the ordered pairs and the vertex.

 

Let's get the vertex:

a is the coefficient of x2 therefore a=1.

b is the coefficient of x: therefore b=1.

c is the numerical part, i.e. the isolated number with no variable part: therefore c = 1..

MATH,

MATH

MATH

Let's get some ordered pairs and then create a graph:

MATH   Figure

ex. 2) Find a,b, and c; the vertex and 4 ordered pairs of the parabola y = -2x2 + 1; graph the ordered pairs.

I can rewrite y = -2x2 + 1 as y = -2x2 + 0x + 1 so I can easily determine a, b, and c.

Let's get the vertex:

a is the coefficient of x2 : therefore a = -2.

b is the coefficient of x: therefore b=0 since we don't have a term with x.

c is the numerical part, i.e. the isolated number with no variable part: therefore c = 1.

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$vertex=(0,1)$

Here are the ordered pairs and the graph of the parabola:

MATH

SECTION 23.2 QUADRATICS AND X AND Y INTERCEPTS

In an earlier chapter, chapter 14 , I reintroduced the concept of x and y intercepts. I will look at the general rules for determining the x and y intercepts of a parabola.


Figure

ex.) Given y = x2 + x - 6

a) Find the x and y intercept
b) Find the vertex
c) Plot the vertex and the x and y intercepts.


ANSWERS:
a) To find the x intercept set y = 0.

$0=x^{2}+x-6$

$0=(x+3)(x-2)$

x + 3 = 0 or x - 2 = 0

-3 and 2 are the x intercepts.

The associated y values are 0.

The associated points are (-3, 0) and (2, 0)

 

To find the y intercept set x = 0.

$y=(0)2+0-6=-6$

-6 is the y intercept and its associated x value is 0.

The associated point is (0, -6)

 

b) y= x + x - 6 = 1x2 + 1x + -6

a = 1: b = 1 ; c = -6

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MATH

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c)

Figure

Also note that since a=1 is positive the graph is concave up.

I will consider an example of a parabolic equation that produces a parabola with no x intercepts:

ex. Determine the x-intercepts for y = x2 - 4x + 9

answer:

y = 1x2 + -4x + 9

I rewrote the original quadratic as addition so I could get a, b, and c.

Therefore a = 1; b = -4; c = 9

To find the x - intercepts, set y = 0, then solve for x: 0 = x2 - 4x + 9

We will use the Quadratic Formula to solve this since we can't factor  x2 - 4x + 9.

MATH

Notice that the radicand is -20 and you can't find the square root of -20. What this reveals is:

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IF YOU GET A RADICAND THAT IS NEGATIVE THEN
YOU CAN CONCLUDE THERE ARE NO X-INTERCEPTS

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ex.) Graph y = x2 - 4x + 9 by getting the vertex and the y intercept and sketch the graph to see if there are no x-intercepts.

Also note that since a = 1 is positive the graph is concave up.

a = 1; b = -4 ; c = 9

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The vertex is: (2, 5)

To get the y-intercept: set x = 0 and solve for y: MATH

Here's the graph:

Figure

 

example) Graph y = -3x2 - 14x - 8 using the vertex and the x and y intercepts.

Note that since a = -3 is negative the graph will be concave down.

MATH

Note that I rewrote the above equation in terms of addition which is the form of the quadratic equation.

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The vertex is: MATH

y-intercept: MATH

x-intercept: MATH

MATH or x + 4 = 0

MATH or x = -4

HERE'S THE GRAPH:

Figure

 

-----------------------------
THE SUMMER STUDY GUIDE
BY CHAPTERS

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RETURN TO THE SUMMER STUDY GUIDE MAIN PAGE

- CHAPTER 1: THE NUMBER SYSTEM

- CHAPTER 2: ORDER OF OPERATIONS

- CHAPTER 3: VARIABLES, MONOMIALS, BINOMIALS, TRINOMIALS, POLYNOMIALS,
COEFFICIENTS, TERMS AND LIKE TERMS

- CHAPTER 4: SIGNED NUMBERS, ABSOLUTE VALUE, AND INEQUALITY SYMBOLS

- CHAPTER 5: FACTORS, COMMON FACTORS, LEAST COMMON FACTORS AND GREATEST COMMON FACTORS

- CHAPTER 6: PROPERTIES OF NUMBERS

- CHAPTER 7: THE WORLD OF FRACTIONS

- CHAPTER 8: EXPONENTS

- CHAPTER 9: ROOTS

- CHAPTER 10: ALGEBRAIC EXPRESSIONS

- CHAPTER 11: CARTESIAN COORDINATE SYSTEM

- CHAPTER 12: SETS, RELATIONS AND FUNCTIONS

- CHAPTER 13: AVERAGE RATE OF CHANGE OF Y WITH RESPECT TO X, SLOPE, PYTHAGOREAN THEOREM, AND DISTANCE FORMULA BETWEEN TWO POINTS

- CHAPTER 14: X-INTERCEPT(ZERO) AND Y INTERCEPT(B)

- CHAPTER 15: LINES

- CHAPTER 16: FUNCTIONS

- CHAPTER 17: MULTIPLYING POLYNOMIALS

- CHAPTER 18: FACTORING

- CHAPTER 19: RATIONAL EXPRESSIONS

- CHAPTER 20: SOLVING EQUATIONS

- CHAPTER 21:SOLVING INEQUALITIES

- CHAPTER 22: SOLVING A SYSTEM OF EQUATIONS

- CHAPTER 23: QUADRATICS

- CHAPTER 24: CIRCLES

- CHAPTER 25: AREAS AND PERIMETERS OF PLANE FIGURES

- CHAPTER 26: VOLUMES