Functions
Complete the following table for y = 5x.
x 0 1 2 3 4 5
y 0 ? ? ? ? ?
x 0 1 2 3 4 5
y 0 5 10 15 20 25
The axis represented by a horizontal line on a coordinate graph.
x-axis
The equation that represents a direct variation.
y = ax where a is a fixed number other than zero.
The general equation for a linear function.
y = ax + b where a and b are constant numbers.
We will learn later that a is the slope of the line and b is the y-intercept (the place where the line intersects the y-axis).
The general equation representing an inverse variation.
y = a/x where a is a constant number other than zero.
Daily Double: The graph of an inverse variation has this shape.
Complete the following table for y = x - 3.
x 3 4 5 6 7
y 0 ? ? ? ?
x 3 4 5 6 7
y 0 1 2 3 4
The axis represented by a vertical line on a coordinate graph.
y-axis
True or False. The distance of a star and the time it takes its light to reach us is a direct variation.
True
Complete the following table for the linear function y = 2x + 1.
x 0 1 2 3 4 5
y ? ? ? ? ? ?
x 0 1 2 3 4 5
y 1 3 5 7 9 11
True or False. A person’s age and the length of his or her attention span is an inverse variation.
False. If this were true, a baby would have a long attention span!
The number of times that the hour hand of a clock goes around the clock is a function of the time. How many times does the hour hand go around the clock in one day (24 hours)?
24
The point at which the axes on a coordinate graph intersect.
The origin, O, or (0,0).
True or False. The weight of a cat and its age is a direct variation.
False
Complete the following table for the linear function y = 2x + 5.
x 0 1 2 3 4 5
y ? ? ? ? ? ?
x 0 1 2 3 4 5
y 5 7 9 11 13 15
True or False. The time required to swim across a lake and the rate at which you swim.
True. If someone swims twice as fast, the time required is cut in half.
The number of times that the hour hand of a clock goes around the clock is a function of the time. How many times does it go around the clock in seven days?
164
The location of a point on a graph is given by this pair of two numbers.
coordinates
The volume of a balloon varies directly the temperature in degrees Kelvin of the air inside it according to the equation y = 2x where x represents the temperature in K and y represents the volume in cubic cm. If the temperature is 300 K, what is the volume?
600 cm3
Mr. Scrooge pays his employees a starting salary of $80 per week with weekly raises of 50 cents. The number of dollars an employee earns in a week, y, is a function of the number of weeks worked, x. Complete the following table for this function.
x 1 2 3 4 5
y 80 80.5 ? ? ?
x 1 2 3 4 5
y 80 80.5 81 81.5 82
The number of times a wheel revolves when rolling one kilometer varies inversely with its diameter. If the wheel’s diameter is measured in meters, the constant of variation for this function is 318. This is the formula for this function when d represents the diameter of the wheel in meters and n represent the number of revolutions it makes in rolling one kilometer.
n = 318 / d
The number of times that the hour hand of a clock goes around the clock is a function of the time. This is the formula for the number of times, n, that the hour hand goes around the clock in d days.
Formula = 24n/d
A function can be represented by pairs of numbers in a table, these pairs of numbers can be used as coordinates of points to make this picture.
Graph
...For coordinate graphs, we use the formula to make a table. And, then we use the table to graph.
The volume of a balloon varies directly the temperature in degrees Kelvin of the air inside it according to the equation y = 2x where x represents the temperature in K and y represents the volume in cubic cm. If the volume is 500 cm3, what is the temperature?
250 K
Mr. Scrooge pays his employees a starting salary of $80 per week with weekly raises of 50 cents. The number of dollars an employee earns in a week, y, is a function of the number of weeks worked, x. Write a formula for this function.
y = 80 + 0.5(x-1)
Or, y = 79.5 + 0.5x
The number of times a wheel revolves when rolling one kilometer varies inversely with its diameter. If the wheel’s diameter is measured in meters, the constant of variation for this function is 318. A wheel with a diameter of 0.5 meter, revolves this many times when rolling one kilometer.
n = 318 / d = 318 / 0.5 = 636 revolutions