Chapter 5 LINEAR FUNCTIONS

Section 5-1 LINEAR FUNCTION – A function whose graph forms a straight line.  Linear functions can describe many real- world situations, such as distances traveled at a constant speed. LINEAR EQUATION – Any equation that can be written in standard form.

STANDARD FORM OF A LINEAR EQUATION Ax + By = C (where A, B, and C are real numbers and A and B are not both 0)  X and y both have exponents of 1.  X and y are not multiplied together.  X and y do not appear in denominators, exponents, or radical signs.

Lesson 5-2  y-intercept: The y-coordinate of the point where the graph intersects the y-axis. The x-coordinate of this point is always 0.  x-intercept: The x-coordinate of the point where the graph intersects the x-axis. The y-coordinate of this point is always 0.

Graphing Ax + By = C Using Intercepts  Find the x-intercept by _______________  Find the y-intercept by________________  Graph the line by____________________

Lesson 5-3  Rate of change – a ratio that compares the amount of change in a dependent variable to the amount of change in an independent variable. change in dependent variable change in independent variable

 Rise – the difference in the y-values of two points on a line.  Run – the difference in the x-values of two points on a line.  slope – The ratio of rise to run for any two points on the line. Slope = rise = change in y run change in x run change in x  The slope of a horizontal line is 0.  The slope of a vertical line is undefined.

Lesson 5-4  You can use the slope formula to find how quickly a quantity is changing.  Slope = change in y change in x change in x Slope Formula  m = y 2 – y 1 x 2 – x 1 x 2 – x 1

Lesson 5-5  Direct Variation – a special type of linear relationship that can be written in the form y = kx y = kx  Constant of variation – “k” the ratio of y/x in an equation with direct variation

Lesson 5-6  Slope-intercept form y = mx + b, Where m is the slope, and b is the y intercept.

Lesson 5-7  Point-Slope Form. y – y 1 = m(x – x 1 )

Forms of Linear Equations STANDARD FORM OF A LINEAR EQUATION Ax + By = C - Useful for identifying linear equations; finding x and y intercepts; and graphing a line using a table of values or x and y intercepts. Slope Formula m = y 2 – y x 2 – x 1 x 2 – x 1 - Used to find the slope of a line when given two ordered pairs.

Direct Variation y = kx (k = y/x is the constant of variation) - Used to graph lines (the constant of variation is the slope of the line) Slope Intercept Form y = mx + b (m = slope, b = y-intercept) -Used to graph a line from an equation or write an equation from two ordered pairs. Point-Slope Form y – y 1 = m(x – x 1 ) - Used to write an equation from two points.

Lesson 5-8  Two different nonvertical lines are Parallel if and only if they have the same slope.  All different vertical lines are parallel.  Two different nonvertical lines are Perpendicular if and only if the product of their slopes is -1.  Vertical lines are perpendicular to horizontal lines.

Lesson 5-9  Transformation – a change in position or size of a figure. Translation of a Linear Function  Translation (slide) – a type of transformation that moves every point the same distance in the same directions When the y-intercept b is changed in the function f(x) = mx + b the graph is translated vertically.

Rotation of a Linear Function  Rotation (turn) – a transformation about a point (the y-intercepts are the same, but the slope is different). When the slope m is changed in the function f(x) = mx + b it causes a rotation of the graph about the point (0,b), which changes the line’s steepness.

Reflection of a Linear Function  Reflection (flip) – a transformation across a line that produces a mirror image. When the slope m is multiplied by -1 in f(x) = mx + b, the graph is reflected across the y-axis.