UNIT 8 Notes
Solving Radical Equations.
Radical Equation: an equation that has a variable in a radicand.
Extraneous Solution: a solution that does NOT satisfy the original equation.
Note:
You must check your answer when solving radical equations.
Solving Rational Equations:
Proportion: two fractions that are equal to each other.
To Solve Rational Equations:
1)Factor the denominators if possible.
2)Find the LCD.
3)Multiply every term in the equation by the LCD.
4)Reduce/clear out the fractions.
5)Solve the equation.
6)Check your answers to make sure the denominators do NOT become zero.
Solving Literal Equations:
Literal Equations: are equations with several different variables.
Transformations of Functions:
How graphs move around
Translation:Change to the Parent Graph:
f (x + h )Translates the graph h units left
f ( x – h )Translates the graph h units right
f ( x ) + kTranslates the graph k units up
f ( x ) – kTranslates the graph k units down
Reflection:Change to the Parent Graph:
-f ( x )Reflects the graph over the x-axis
f ( - x )Reflects the graph over the y-axis
Dilation:Change to the Parent Graph:
a * f ( x ), a > 1Stretches the Graph Vertically
a * f ( x ), 0 < x < 1Compresses the Graph Vertically
Graphing Radical Functions.
Radical Functions: a function that contains an
x in the radicand.
To Find the Domain of a Square Root Function:
Set the radicand ≥ 0 and solve.
Graphing Rational Functions.
Rational Function: a function that contains an x of degree one or higher in the denominator.
Asymptotes: horizontal and vertical lines that guide the graph of a rational function.
To find the Horizontal Asymptote:
y = the number to the right of the fraction.
Example: y = 2 + 4 y = 4 is the horizontal
x – 2 Asymptote.
To find the Vertical Asymptote:
Set the denominator equal to zero and solve.
Example: y = 2 + 4 x – 2 = 0
x – 2 x = 2 is the vertical Asymptote.
Domain: All Real Numbers except the restriction.
Range: All Real Numbers except the Horizontal
Asymptote
Inverse Variation.
Direct Variation: a linear function in the form of
y = kx .
A direct variation is a linear function.
The graph of a direct variation always passes through the origin.
k → constant of variation = slope.
k = y/x
Inverse Variation: an equation in the form of
xy = k or y = k/x
k → constant of variation
Solving Combined Variation Problems:
combination of direct variation (y = kx)
inverse variation
and
joint variation (y = kxz) equations.
When dealing with word problems, you should consider using variables other than x, y, and z, you should use variables that are relevant to the problem being solved. Also read the problem carefully to determine if there are any other changes in the combined variation equation, such as squares, cubes, or square roots.
Step 2: / Use the information given in the problem to find the value of k, called the constant of variation or the constant of proportionality.
Step 3: / Rewrite the equation from step 1 substituting in the value of k found in step 2.
Step 4: / Use the equation found in step 3 and the remaining information given in the problem to answer the question asked. When solving word problems, remember to include units in your final answer.
Combined Formulas:
Y varies directly with x and inversely with z → y = kx/z
Y varies jointly as x and z and Inversely as d → y = kxz/d
Piecewise Functions:
A Piecewise is a function defined by at least two equations ("pieces"), each of which applies to a different part of the domain.
Functions
1)Linear Functions – highest power of x is 1. They form a straight line on the graph.
The equations are in the form of y = mx + b.
The equation of a horizontal line is in the form of y = the y-intercept.
The equation a vertical is in the form of x = the x – intercept.
A VERTICAL LINE is NOT a Function.
2)Quadratic Functions – the highest power of x is 2. They form a parabola that opens upward or downward. The equations are in the form of:
y = ax²
y = ax² + c
y = ax² + bx + c
3)Absolute Value Functions - the variable “x” is contained inside the Absolute Value Symbols.
The graph forms a V – shape that opens upward or downward.
4)Exponential Functions – are in the form of y = a · bx .
The graph forms a curve upward or downward.
5)Square Root Functions – otherwise known Radical Functions – the variable “x” is contained in the radicand.
The graph forms a curve to the right.
6)Rational Functions – contains the variable “x” in the denominator.
The graph forms 2 curves and there are asymptotes that guide the graph.
To Find the Horizontal Asymptote: y = the number to the right of the fraction.
To find the Vertical Asymptote: Set the denominator equal to zero and solve the equation for x.
Parent Functions:
are the simplest form of that type of function, meaning they are as close as they can get to the origin (0, 0).