Chapter 10 Math Formulas and example problems
Find the axis of symmetry:
(x=-b/2a)
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
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 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 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}.
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.
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.
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).
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.
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
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}
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.
