Chapter 9

Describe the rule for the parabola getting narrower.

as the absolute value of a gets larger (not the value of a itself, but its distance from 0) the parabola gets narrower

Why does it make sense for the axis of symmetry to be an x= equation?

because it is a vertical line through the parabola

How many points do you need when solving a quadratic equation by graphing?

enough points to pass through the x axis

What is the first step to making a table of values?

finding the axis of symmetry

What are the steps to graphing a quadratic function?

finding the x coordinate of the vertex, making a table of values using x values to the left and right of the vertex, and plotting the points and connecting them with a smooth curve to form a parabola

Give the falling object model formula and what each letter stands for.

h=-16t²+s, h is height in feet, t is time in seconds, s is initial height, the s value changes depending on the height

Give all of the possibilities for the discriminant and what they indicate about the amount of solutions.

if D is positive there are 2 real roots, if D is negative there are no real root, if D is 0 there is one real root

Describe what each possibility for D indicates about the parabola.

if D is positive, the parabola hits the x axis in two places, it crosses the axis twice because the answers are x intercepts and there are two answers, if D=0, the vertex is on the x axis so it hits the x axis once and there is one x intercept and therefore one solution, if d is negative the parabola never hits the x axis and there are no x intercepts, no solutions

Square root rule and example

if b²=a, then b is a square root of a, if 8²=64, 8 is a square root of 64

Will the x intercept always be a whole number?

no, it can be a fraction, sometimes you have to approximate

When you simplify and get a perfect square, is the number that is the square root in the √ symbol? Ex: √75 is simplified to √25 and √3, is √25 simplified to √5 or 5?

no, it is not in the √ symbol, it is just simplified to 5

Is the coordinate written in the final answer?

no, it is only part of finding the final answer, the final answer is just x=the x intercepts

Is there always symmetry in the table?

no, the vertex might not be in the middle so the chart might not seem symmetrical

Explain how many different answers there can be and when each occurs.

one answer if the vertex is on the x axis, no answer if it never hits the x axis, two answers if there are two x intercepts

Explain how to solve a question involving perimeter of a rectangle.

one side is x and one is y but you have to get one variable, so one of the values becomes the potential perimeter minus the other variable, then set it up for the potential area to equal the variable times (potential perimeter minus the variable) then put it into standard form and find the discriminant, if it is positive then the given perimeter and area is possible

What is the axis of symmetry?

the imaginary line passing through the vertex on every parabola that divides the parabola into two symmetric parts which are mirror images of each other

What is a vertex?

the lowest point of a parabola that opens up or the highest point of a parabola that opens down, it is the turnaround point, every parabola has a vertex

What if the negative is outside of the principal square root sign?

the negative is an afterthought and it is just added on, therefore the answer would be a negative

What is the solution to an equation?

the places where 0 shows up on the graph as a y-value, the roots of the quadratic equation are the x-intercepts of a parabola, the places where y=0, it is written as x=, separately for each number

When does h equal 0 in the falling object model?

when the object hits the ground

What is the defining part of a quadratic equation?

x must be squared - all quadratic equations have x² and no higher than squared

Does the sign with each coefficient in the equation stay with it in the quadratic formula?

yes - if it is minus a coefficient in the equation, it becomes a negative when used in the quadratic formula

If the discriminant is 0, will it be possible?

yes because there is one answer, which means there is a solution, so it is possible

What if there is a negative in the numerator and the denominator in the simplified version of a simplest radical form solution?

you can get rid of both negatives because the square root is ± so dividing it by a negative doesn't change anything, canceling out both negatives is the simplest form

Explain how the product property can work in reverse.

you can multiply square roots together to get a perfect square, the rule goes in both directions - if they are multiplied by each other you can put them back together as well as take it apart - for multiplication, not addition

Explain when you can rearrange the equation and when you can't.

you can rearrange it for solving, you cannot when you are told to graph a specific equation

Explain simplifying with reducing fractions with a radical.

you can reduce fractions - simplify like in a linear equation - but you do not change the radical

How can you tell if the parabola goes up or down? How can you tell if it is wider or narrower?

you can tell if it goes up or down by if it is positive or negative, you can tell if it is wider or narrower by its absolute value

What do you do with a fraction under the radical sign?

you cannot leave a fraction under the radical sign, you can take apart fractions or put them together then divide

What do you do with radicals in the denominator?

you cannot leave radicals in the denominator, so multiply them by the radical over the radical which is like multiplying by one, it works because multiplying the square root of a number by the square root of a number gives you the number

What if the numerator and denominator can be simplified but the coefficient of a radical cannot be?

you cannot simplify further because even if there is no coefficient, there is the implied coefficient 1 which cannot be simplified and you must simplify all 3 terms

How do you know if something in a question is possible?

you know this by determining if it is solvable, and it is solvable if the discriminant is positive, if the discriminant is negative a problem is not solvable

When asked as the square root and not specifically the principal square root of 64 what is the answer?

±8

If d is negative, what is the answer?

ø (no real solution)

Explain the quotient property of radicals.

√a/b =√a/√b, b must be positive because it cannot be a negative square root and it cannot be 0 because 0 cannot be in the denominator, a can be greater than or equal to 0 (positive number or 0)

Explain the product property of radicals.

√ab=√a+√b, the square root of a product can be broken up into the square root of each part of the product but a and b must be positive, they cannot be negative because you cannot get the square root of negative numbers - does not apply for addition, you cannot split up addends

What are the steps to solving a quadratic equation using the quadratic formula?

1) put the equation in standard form 2) identify the coefficients 3) sub in the coefficients and simplify 4) when only ± is left, split into two separate equations, one with x+ and one with x-, and when simplified these two numbers are the final solutions

How many solutions can there be if d is positive?

2

If you have a square root of the sum of two numbers, can you split up the numbers and do the square roots separately? Ex) √(9+16) - does it make sense to do the square root of 9 plus the square root of 16?

No, you cannot do that because √9+√16 does not equal 5, the square root of √(9+16), and since it doesn't work for numbers it also doesn't work for variables - you cannot separate the square root of a sum or difference

Why does finding the x axis work as a solution?

because finding the x axis is when y is set to 0, this is what happens when you graph the standard form of the quadratic equation

Why does solving by graphing a quadratic equation work?

because the solutions are the x intercepts

What is an irrational number? What is a rational number? Give examples of irrational numbers.

a number that cannot be written as a quotient of two integers - cannot be written as a fraction or as a decimal that stops, it can only be an approximation on the calculator (π, √2, √3, √5, ₃√2, √6, √7, √8), rational numbers can be written as a fraction

What is a perfect square? Give some examples.

a number whose square root is an integer or quotient of integers - when a number that is an integer or an integer over an integer is squared, an integer times itself (121, 49, 100/9, 25/36)

Explain what the graph of a quadratic function looks like.

all quadratic functions have U shaped graphs called parabolas, if a, the leading coefficient, is positive, the parabola opens up, if a is negative, it opens down and forms an upside down U

What is a quadratic equation?

an equation which can be written in the standard form ax²+bx+c=0 (a≠0) - a, b, and c are coefficients/constants

What is the difference between undefined and no solution?

an expression is undefined, an equation has no solution

What is a radical expression?

an expression that has square/cube/other types of roots in it

What should you do when there are fractions as a, b, or c?

clear the fractions so a, b, and c are whole numbers so you don't have to deal with square roots of fractions

What should you do if you need to determine a value for c for a given amount of solutions?

create an inequality for the given number of solutions, if it is one solution then it will be an equation not an inequality - to set up the inequality set up the discriminant value then greater than 0, solve the inequality to find the cutoff point

What are the three possible cases for quadratic equations? What is the maximum number of answers for quadratic equations, and what is the rule for maximum number in general?

d is positive, d is 0, d is negative, a maximum of 2 solutions, the maximum number of possible solutions per equation is equal to the exponent sign on the variable

In a word problem, how do you find maximum height?

find the y coordinate of the vertex

What topics do questions involving the discriminant relate to?

if it can be done or how many answers are there, in order to decide when you should use the discriminant you have to decide what is being asked

When can a solution be left with ±?

if it is in simplest radical form or if the two answers are the same absolute value but one is positive and the other is negative

Explain how to simplify radicals according to the product property.

if there is a part of the product that is a perfect square, you have to break it down to have a perfect square - if there is no perfect square it is irrational and left as a radical - simplify until there is no perfect square inside √ as a factor

How are irrational answers written? How are perfect square (rational) answers written?

irrational answers are either written with ± and the √ symbol or as an approximated decimal depending on what is asked, but perfect squares are simplified to be a number, not with the √ symbol

What is the principal square root? Give an example.

it is a number in the symbol √ and cannot be a negative number. The principal square root of 64 is 8, not -8. Basically principal square root is the positive part of the square root.

What does a graph of a quadratic equation with a negative coefficient look like? What would it look like if the coefficient is positive?

it will be an upside down U if it is negative and a normal U if it is positive

What is the purpose of the quadratic formula, and what does it use from the quadratic equation?

its purpose is to solve for x using the coefficients a, b, and c

What should you look for when simplifying and when should you stop?

look for perfect squares and keep going until no more perfect squares are factors and the factors cannot be broken down into any more perfect squares

Describe the backup method.

make a factor tree (list two factors then keep going and circle all prime numbers, put prime numbers in square root symbol) if a number appears twice then put one number from each pair of the same number out of the √ symbol and keep numbers that only appear once in the √ symbol

Can you clear the fractions for a problem relating to the discriminant?

no

Do all numbers have a real square root? Why?

no because negative numbers do not have a real square root

How do you check when solving by graphing a quadratic equation ?

plug each x intercept into the x value of the equation

Should the answer be the positive or the negative for a falling object model? Should you give a decimal or simplest radical form?

positive because you are finding the value of t, which is time, and you cannot have negative time, you should give a decimal answer to show how big the number is

When making a table, what should you do with the vertex?

put it in the middle and then find y values for the same number of x values on each side of the vertex

To solve an equation by graphing, what are the steps?

put it into standard form with =0 on one side, then change this to a y= equation, go through the steps to graph the parabola, the solutions are the x intercepts

Explain how to put the solution in simplest form.

reduce the fractions, divide all of the terms by the same number but do not divide the square root, all terms are divided but the square root symbol is not because it is part of another term, however you simplify the square root until there are no perfect squares left inside the radical

Explain when you would use decimal form and when you would use radical form. What can be left in a simplest radical form solution?

simplest radical form can be left with the ± sign and the radical sign in the form of a fraction, but word problems often simplify the radical to a decimal to the give the feel of the size of the number in the answer

Explain the difference between velocity and speed.

speed is an absolute value but velocity has direction so it can be positive or negative depending on which way it is going

What should you do after finding out the coordinates of the vertex?

sub this information into the equation and put the resulting points in the table

What is the discriminant and what does it do?

the discriminant, represented by D, is D=b²-4ac, it helps to determine the nature of the roots, it is the part of the quadratic formula under the square root symbol?

What is a parabola? Does it have to be a graph of a function? What does it look like when it is a function?

the graph of a quadratic relationship, it does not have to be a graph of a function but in Algebra I it is, when it is a function it is either a U or an upside down U

Describe what happens as a, the leading coefficient, gets larger.

the parabola gets narrower and steeper

What are the roots of a quadratic equation?

the solutions of ax²+bx+c=0 - roots is a fancy word for solution

What are the coordinates/equations of the vertex and the axis of symmetry?

the vertex has the coordinate -b/2a and the axis of symmetry is the vertical line x=-b/2a

What would happen if a was 0 in a quadratic equation?

the x² part would drop out of the equation and it would no longer be quadratic

How many answers are there if d=0? What are they?

there is only one answer: x=0

What is the principal square root of a negative number?

undefined - if the negative is inside the square root symbol it is undefined

Explain the model for when an object is thrown.

when an object falls, the only force is gravitational pull, but when it is thrown there is also the initial force from the throw, so the model is h=-16t²+vt+5, v is initial velocity, h is height in feet, t is time in seconds, s is initial height

What is the quadratic formula?

x=-(b±√b²-4ac)/2a

Give the axis of symmetry formula, the quadratic formula, and how they are used to get the vertex.

x=-b/2a, the quadratic formula is y=ax²+bx+c, the b value from the quadratic formula is the numerator over 2 times the a value, find the result and this is what x equals, then sub it back into the quadratic formula to get the y value, this coordinate set is the vertex

What is the format for the solution of a quadratic equation x²=d if d is positive?

x=±√d

Give the standard form of the quadratic function.

y=ax²+bx+c (a≠0, a is positive)

Can a radical be left in the numerator?

yes