Linear Functions Linear Algebra.

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Presentation transcript:

Linear Functions Linear Algebra

A Linear Function is… From a Graph: A linear function is a function (or rule) that has a constant rate of change From a Graph: Linear Functions will graph as one straight line

A Linear Function is… From an Equation: A linear function is a function (or rule) that has a constant rate of change From an Equation: The equation will not have exponents greater than 1 on the variables The equation will not have a variable in the denominator of a fraction Examples of linear functions: y = 2x + 1 y = -1/2x – 5 y = -4x – 2.8 Examples of non-linear functions: y = 3x2 + 2x + 5 y = -5x3 y = 8/x

A Linear Function is… From a Table: A linear function is a function (or rule) that has a constant rate of change From a Table: The rate of change will be the same every time Constant rate of change because the ys change by +12 for each change in 1 x Constant rate of change because the ys change by +4 for each change in 1 x +1 +12 +4 +1 +1 +12 +1 +4 +.5 +6 It’s even ok here because it went up by 6 for .5 of an x, therefore will go up a full 12 for a full 1 x +1 +4

Speeding Rabbit From a Word Problem: D = rt Suppose a rabbit travels 256 feet in 4 seconds. Assume that this is a constant speed. Situations with a constant rate of change will be linear functions. Write a linear equation in two variables to represent the situation. In science you learned the relationship between distance and time. What is that equation? D = rt What values were given in this story that we could substitute into this formula? Distance = 256 feet Time = 4 seconds We can solve for “r” rate.

D = rt 256 = r(4) Plug in the values we know for D and r 256 = r(4) Solve the equation by dividing both 4 4 sides by 4. 64 = r r = 64 feet/second (that is his rate/constant speed) The rabbit’s personal equation to determine how far he could go in a certain amount of time is: d = 64t.

Use the equation to make predictions about the distance the rabbit traveled over various intervals of time. How can an equation help make predictions about distance the rabbit traveled in various amounts of time? We could make a table of values. The x column would correspond to time the rabbit ran (the input). The y column would correspond to the distance the rabbit ran AFTER the time had passed (the output). We could substitute (plug in) the times into the rabbit’s formula D = 64t. Time (t) Distance (D) 64 1 2 128 3 192 4 256 5 320 The equation (function) allows us to PREDICT the distance the rabbit has traveled after any duration of time.

A Linear Function is… Remember a function is a rule that assigns each input exactly one output. A rule is an equation. The equation for the rabbit’s distance after a specific amount of time was d = 64t. That equation is a function. Each input (time) had exactly one output (distance). This means at 3 seconds the rabbit had to have traveled 192 feet. The answer couldn’t have been 180, 192 and 300 feet. At 3 seconds the rabbit traveled 192 feet. There is only one possible output. The rabbit’s function is a LINEAR FUNCTION because he traveled at a constant rate of change (or constant speed not getting faster or slowing down)

Walking Race: Mr. Furman and Mr. Ganey are in a walking race. Mr. Furman walks 18 meters in 6 seconds and Mr. Ganey walks 24 meters in 6.25 seconds. Find each teacher’s walking rate. D = rt Plug in the values we know for each and solve for the unknown r (which will be the rate or speed) Mr. Furman: 18 = r(6) Mr. Furman’s walking rate is 3 meters/second. Mr. Ganey: 24 = r(6.25) Mr. Ganey’s walking rate is 24/7 or 3.84 meters/second. Who will win the walking race?

Walking Race: Write a linear function to represent each teacher’s distance after any amount of time. Mr. Furman: D = 3t Mr. Ganey: D = 3.84t How long will it take Mr. Furman and Mr. Ganey to speed walk 150 meters? Substitute 150 meters into each equation and solve for time. Mr. Furman 150 = 3t 3 3 50 = t It will take Mr. Furman 50 seconds to speed walk 150 meters. Mr. Ganey 150 = 3.84t 3.84 3.84 39.06 = t It will take Mr. Ganey 39 seconds to speed walk 150 meters.