Chapter 10 Math Formulas and example problems

Find the axis of symmetry:

(x=-b/2a)

Solve x^2+9x+18?

1.) Equation already in standard form and is ready to be factored setting each solution to "0". 2.) Find the product of 18 and the sum of 9. Answer is (x+6) (x+3) =0 Now, set each equation to 0. x+6=0 and x+3=0. Now, subtract a -6 from 6 and a negative 6 from 0. You get x=-6 NOW do the same to the other side. X+3=0 subtract -3 from 3 and -3 from 0. You get a -3. So, the solution to the set is {-6,-3}. It is in this format because -6 is smaller than a -3. Numbers are as such -6,-5,-4, -3 ,-2, -1, 0,1,2,3,4. since the -6 is farthest out on the number line it is the smaller of the two numbers.

What is a constant function?

Constant function is a linear function of the form y = b, where b is a constant. It is also written as f(x) = b. The graph of a constant function is a horizontal line. Whose y-intercept is (0,b).

Examples of this

Domain: [-4,infinity sign) range: [-3, infinity sign) x-intercept (5,0) y-intercept (0,-1)

Example of a graph that is not a function.

See attachment

Graph f(x) = sq. rt. sym over x+2 -7

Shift the basic graph of f(x) =sq.rt.sym over x left by 2 units and down 7 units h=-2 and k=-7 Domain:[-2, infinity) range: [-7,inf. sgn.)

Graphing a square root function

Shift the graph of f(x)= square root over x by h units horizontally and by k units vertically. f(x)=a square root symbol over x-h +k

Solving equations using a u-substitution

Solve x^4 -5x^2-36=0 x^4-5x^2-36=0 Let u=x^2 then u^2-5u-36=0 factor (u+4) (u-9)=0 setting to "0" u+4=0 and u+9=0 u=-4 and u=9 then square x^2=-4 and x^2=9 then square root symbol over x^2 =+/-square root symbol over -4 x=+/- 2i same for 9 x= +/- 3 {-2i,2i,-3,-3}

Formula for vertex:

x=-b/2a

How do you solve a quadratic equation by factoring?

1. Write equation in standard form ax^2 +bx+c=0 2. Completely factor the quadratic expression. 3. Set each factor equal to 0 and solve.

x^2-50=0

1.) Add 50 to both sides of the equation to isolate x^2. x^2=50 2.) Take the square root of each side square root of x^2 = x. Square root of 50 is 5 square root of 2. 3.) Simplify the square root and set both terms equal to (+-). The solution to this set is {-5 square root over 2, 5 square root over 2}.

How do you determine a linear function from its graph?

1.Determine the coordinates of two points (x 1,y1) and (x2,y2) on the line. 2. Find the slope m on the line using the formula: m=y2-y1/x2-x1 3.) Set f(x) = y2 and solve for b. 4. Write the formula for the function using f(x) =mx+b.

Example

2x^2+9x+11=0 formula b^2-4ac=9^2-4(2)(11)=81-88=-7 because the discriminant is (-) is has two nonreal solutions.

What is a function?

A function is a rule that takes an input value and assigns a particular ouput to it. In mathematics, a function[1] is a relation between a set of inputs and a set of potential outputs with the property that each input is related to exactly one output. An example of such a relation is defined by the rule f(x) = x2, which relates an input x to its square, which are both real numbers. The output of the function f corresponding to an input x is denoted by f(x) (read "f of x"). If the input is -3, then the output is 9, and we may write f(-3) = 9.

What is a linear function?

A linear function is a function that can be written in the form f(x)=mx+b, where m and b are real numbers.

What can a quadratic formula do to a standard form equation?

A quadratic formula can be used to solve any quadratic equation that is written in standard form. ax^2+bx+c=0

Quadratic functions:

A quadratic function is a function that can be written in the form f(x) =ax^2 +bx+c, where a, b, and c are real numbers a does not equal to 0.

What is the vertical line test?

A test use to determine if a relation is a function. A relation is a function if there are no vertical lines that intersect the graph at more than one point. In mathematics, the vertical line test is a test to determine if a curve is a relation or graph of a function when the function's domain and range correspond to the x and y axes of the Cartesian coordinate system. As a relation or graph of a function can only have one output for each unique input, such a Cartesian representation of the function can have at most a single y value for each x value. Thus, a vertical line drawn at any x position on the graph of a function will intersect the graph at most once.

Graphing the functions of the form

Determine whether the parabola opens upward or downward. Formula f(x) =a(x-h)^2+k graph f(x) =(x+3)^2-5 Opens upward h=-3k k=-5 vertex: (-3,5) y intercepts (0,4) then plug in numbers from vertex to solve for x intercepts.

Example of an evaluation of a function.

For f(x) =x^2+7x-24, find f (-5). f(-5)= (-5)^2 +7 (-5) -24 = 25-35-24=-34

What is function notation?

Function notation is a way to present the formula for the output value of a function for the input x.

Example of this:

Graph f(x) =-2/3 x+5 y-intercept: (0,5) slope:-2/3 on the graph down two and right 3.

Inverse of a function

If f(x) is a one to one function and f(3)=10 and then f -1(10) =3

How do you interpret a graph?

The domain of a function can be read from Left to right along the x-axis on a graph. The range of a function can be read vertically from the bottom to the top of the graph along the y-axis. An x-intercept is a point at which the graph intersects x-axis. The y coordinate of an x-intercept is 0. A y-intercept is a point at which a graph intersects the y-axis. The x coordinate of a y-intercept is 0.

Formula for an area of a rectangle

The formula for the area of a rectangle is Area=Length x width.

Graph f(x)=-6

The graph would be at (0,-6) and it would be a horizontal line. ___________________

What is the input to a function?

The input to a function is often called the argument and the output is often called the value.

What is the formula for a period of a pendulum?

The period of a pendulum is T = 2 pie square root L/32

What is the quadratic formula?

The quadratic formula is x= -b +- square root over b^2-4ac all over 2a.

What is a discriminant?

The radicand b^2-4ac is called the discriminant. If the discriminant is (+) the quadratic equations has two real solutions. The same for (-). If the discriminant is equal to "0" it has one real solution.

What is the standard form of an equation?

The standard form of an equation is ax^2+bx+c=0

How do you evaluate a function?

To evaluate a function for a particular value of the variable, substitute that value for the variable in the function's formula, then simplify the resulting expression.

Solving radical equations

To solve radical equations: 1. Isolate the radical (or one of the radicals) to one side of the equal sign. 2. If the radical is a square root, square each side of the equation. (If the radical is not a square root, raise each side to a power equal to the index of the root.) 3. Solve the resulting equation. 4. Check your answer(s) to avoid extraneous roots.

Example The length of a rectangle is 3 inches more than its width. If the area of the rectangle is 100 inches find its two dimensions. Round to the nearest inch.

Unknowns: Length x+3 Width: x x(x+3)=100 x^2+3x-100=0 then put into a quadratic equation.

Verifying 2 inverse funct.

Verify that f(x) =2x-3 and g(x) x+3/2 are inverse funt. (f x g) (x) = f(x+3/2) then solve 2 (x+3/2)-3 x+2-3 =x Now for (g x f) (x) g (2x-3) (2x-3)+3/2 =2x?2=x The functions are inverse.

How do you graph linear functions?

You can graph a linear function of the form f(x) =mx+b, where m and b are real numbers. To graph a linear function, begin by plotting its y intercept at (0,b). Then use the slope m to find other points.

How do you solve quadratic equations by extracting square roots?

You can solve quadratic equations by extracting square roots by: 1.) Isolate the squared term. 2.) Take the square root of each side. (REMEMBER TO TAKE THE POSITIVE AND NEGATIVE (+-) SQUARE ROOT OF THE CONSTANT.) 3.) Simplify the square root. 4.) Solve by isolating the variable.

When can you use a quadratic formula?

You can use the quadratic formula when the quadratic equation is in standard form and it cannot be solved with any other method.

What is the quotient funct.

divide f(x)/g(x) does not equal 0 the equal sign has a line thru it..

Example of a function notation?

f(x) = 3x+14 Function name: f and input variable: x Formula for output: 3x+14

An example of this looks like this:

f(x) =7x-18

Example

f(x) =x^2-7x+18

What is the product funct.

for any two funct. same as above (f)(x) and g(x).

What is the difference function?

for any two functns. f(x) and g(x) (f-g)(x) =f(x)-g(x)

What is the formula for projectile problems?

h(t) =-16t^2+vot+s where vo is the initial velocity of the projectile and and s is the initial height. vo=80 and s=240 h(t) =-16t+80t+240 h(t)=0 -16t^2+80t+240=0 factor -16 (t^2-5t-15)=0 t= put into quadratic equation formula and solve. Omit negative solution. Round to the nearest tenth of a second.

Example of this:

m= -3-(-5)/6-3 =2/3 f(x) =2/3x+b f(x) 2/3x -7 f(3) =-5 2/3(3) +b =-5 2+b+-5 b=-7

Solving quadratic inequalities

solve inequality by factoring setting to 0 put on a number line and add test points answers that are solutions have a clear circle on them for test points -3 next point -4 put it into test point box and solve for the resulting equation x^2-6x-27>0 -4 test point true, 0 test point false and 10 true.

x^2=28

square root of x^2 =(+-) square root of 28 1.) Take the square root of each side. 2.) Simplify the square root x=(+-) the solution to the set is {-2 square root over 7, 2 square root over 7}

example

square root sign over 4x+33 +3 is not underneath the square root symbol =x to solve subtract 3 from x looks like x-3 on the side of the problem then solve as if you are extracting square roots the term that was squared now looks like 4x+33=(x-3)^2 solve as you would any other equation. 4x+33=x^2-6x+9 then it gets set to "0" x^2-10x-24 0=(x+2) (x-12) x+2=0 and x-12=0 x=-2 and x=12 omit negative solution

Example

x^2+6x-20=0 a=1 (x^2) b=6 c=-20 then substitute into the formula. -b+/-square root symbol over b^2-4ac/2a x=-6+/- square root symbol all over 6^2 -4(1)(-20)/2(1)

Solving rational inequalities

x^2-2x-35/x^2-4 >/0 set each problem top one to zero and bottom to 0 factor out each one then each equation should give us four points and put on a number line along with the U symbol and the infinity sign. If it is a part of a solution the bracket looks like this ] for example 5] or [7 those circles are darkened. anything that is not a solution has parantheses. Make test points and put into a box.

How do you solve quadratic equations by factoring?

x^2-7x=-10 1.) Add 10 on the left side to collect all like terms. Looks like this x^2-7x+10=0 2.) Factor (x-2) (x-5) = 0 3.) Set each factor equal to 0 Case 1: (x-2) =0 Case 2: (x-5) =0 then solve the resulting equations x-2=0 x=2 and x=5. The solution to the set is {2,5}.