Alg. 2, 2-2: Linear Relations and Functions

Alg. 2, 2-2: Linear Relations and Functions

by Jeremy Adelmann

Multiple Select

Vocabulary:  Please Take Notes

Linear Relations:  relations that have straight line graphs

• Linear Relations:  relations that have straight line graphs

• Linear Relations:  relations that have straight line graphs

Multiple Select

Vocabulary:  Please Take Notes

Linear Function: a function with ordered pairs that satisfy a linear equation. Can be written in the form f(x)=mx+b where m and b are real numbers.

• Linear Function: a function with ordered pairs that satisfy a linear equation. Can be written in the form f(x)=mx+b where m and b are real numbers.

• Linear Function: a function with ordered pairs that satisfy a linear equation. Can be written in the form f(x)=mx+b where m and b are real numbers.

Multiple Select

Vocabulary:  Please Take Notes

Nonlinear relations: relations that are not linear

• Nonlinear relations: relations that are not linear

• Nonlinear relations: relations that are not linear

Multiple Select

Vocabulary:  Please Take Notes

Linear equation: an equation with no other operations other than addition, subtraction, and multiplication of a variable by a constant. Variables CANNOT be multiplied together or appear in the denominator and cannot have exponents other than 1. The graph is always a line.

• Linear equation: an equation with no other operations other than addition, subtraction, and multiplication of a variable by a constant. Variables CANNOT be multiplied together or appear in the denominator and cannot have exponents other than 1. The graph is always a line.

• Linear equation: an equation with no other operations other than addition, subtraction, and multiplication of a variable by a constant. Variables CANNOT be multiplied together or appear in the denominator and cannot have exponents other than 1. The graph is always a line.

​Identify Linear Functions

​State whether each function is a linear function.

​

A.

​

• ​This is a function because because it can be rewtitten into the slope intercept form of a line, f(x) = mx + b.

​Identify Linear Functions

​State whether each function is a linear function.

​

B.

​

• ​This is NOT a function because because it cannot be rewtitten into the slope intercept form of a line, f(x) = mx + b.

• ​It cannot be a function when the variable is the denominator of a fraction.

​Identify Linear Functions

​State whether each function is a linear function.

​

C.

​

• ​This is NOT a function because because it cannot be rewtitten into the slope intercept form of a line, f(x) = mx + b.

• ​It cannot be a function when there are more than one variable.

Multiple Choice

State whether each equation or function is a linear function.  Explain.

$$3y-4x=20$$  

• Yes; it can written in the form  $$f\left(x\right)=mx+b$$  .

• No; the variable is in the denominator of a fraction.

• No; it has exponent other than 1.

• No; the variable is under a square root.

Multiple Choice

State whether each equation or function is a linear function.  Explain.

$$y=x^2-6$$  

• Yes; it can written in the form  $$f\left(x\right)=mx+b$$  .

• No; the variable is in the denominator of a fraction.

• No; it has exponent other than 1.

• No; the variable is under a square root.

Multiple Choice

State whether each equation or function is a linear function.  Explain.

$$h\left(x\right)\ =6$$  

• Yes; it can written in the form  $$f\left(x\right)=mx+b$$  .

• No; the variable is in the denominator of a fraction.

• No; it has exponent other than 1.

• No; the variable is under a square root.

Multiple Choice

State whether each equation or function is a linear function.  Explain.

$$j\left(x\right)=2x^2+4x\ +\ 1$$  

• Yes; it can written in the form  $$f\left(x\right)=mx+b$$  .

• No; the variable is in the denominator of a fraction.

• No; it has exponent other than 1.

• No; the variable is under a square root.

Multiple Choice

State whether each equation or function is a linear function.  Explain.

$$g\left(x\right)=5+\frac{6}{x}$$  

• Yes; it can written in the form  $$f\left(x\right)=mx+b$$  .

• No; the variable is in the denominator of a fraction.

• No; it has exponent other than 1.

• No; the variable is under a square root.

Multiple Choice

State whether each equation or function is a linear function.  Explain.

$$f\left(x\right)=\sqrt[]{7+x}$$  

• Yes; it can written in the form  $$f\left(x\right)=mx+b$$  .

• No; the variable is in the denominator of a fraction.

• No; it has exponent other than 1.

• No; the variable is under a square root.

​Standard Form of a Linear Equation

The standard form of a linear equation is Ax + By = C​, when...

• ​A, B and C are intergers with a GCF of 1,

• ​A is greater than or equal to zero, and

• ​A and B cannot both be zero.

​

​Example:

​​Standard Form of a Linear Equation

​Write each equation in standard form.

​

​

Subtract 4x from both sides.

​Combine like terms.

​​Standard Form of a Linear Equation

​Write each equation in standard form.

​

• Subtract 4x from both sides.

• ​Combine like terms.

• ​Put into correct order

• ​Multiply both side by -1 to

make the A value positive

• ​In Standard Form

​​Standard Form of a Linear Equation

​Write each equation in standard form.

​

• Add 9 to both sides.

• ​Combine like terms.

​

• ​Divide both side by GCF, 3.

• ​In Standard Form

Fill in the Blanks

Wrtie the equation in standard from.

$y\ =\ -8x\ -5$  

Fill in the Blanks

Wrtie the equation in standard from.

$5y\ =\ 15x\ -\ 90$  

Fill in the Blanks

Wrtie the equation in standard from.

$\frac{2}{3}y\ -\ \frac{3}{4}x\ +\ \frac{1}{6}\ =\ 0$  

​x - and y - intercepts

​X-intercept:

• The x coordinate of the point at which the graph crosses the x-axis

• ​The y coordinate will be zero. (x, 0)

• ​

Y-intercept:

• The y coordinate of the point at which the graph crosses the y-axis

• ​The x coordinate will be zero. (0, y)

​Find the x and y Intercepts

​Find the x- and y-intercepts of the graph 2x - 3y + 12 = 0.

​

​Start by puttingin standard form.

• ​Subtract 12 from both sides.

• ​Simplify

​Find the x and y Intercepts

​Find the x- and y-intercepts of the graph 2x - 3y + 12 = 0.

​​

​A. Find the x-intercept

​

​Let y = 0

​Multiply

​Simplify

​

​Divide both sides by 2

​The x-intercept is -6

​Find the x and y Intercepts

​Find the x- and y-intercepts of the graph 2x - 3y + 12 = 0.

​​

​B. Find the y-intercept

​

​Let x = 0

​Multiply

​Simplify

​

​Divide both sides by -3

​

​The y-intercept is 4

Fill in the Blanks

Find the x-intercept of the graph of the equation.

$5y-15x=-90$  

Fill in the Blanks

Find the y-intercept of the graph of the equation.

$5y-15x=-90$