Functions
Mathematics, Grade 8
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Study Guide Functions Mathematics, Grade 8
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FUNCTIONS What Are Functions? A function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are performed on the input. • An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the output, y. By substituting a number for x, an output can be determined. • A table can also be used to show input and output values. In this situation, the rule must be determined. A linear function is a function that represents a straight line. A linear function does not have to be in the form of y = mx + b, such as the linear function, 3x + y = 8. However, in order to graph a linear function, the equation should be put into the form y = mx + b, which is the equation of a line. • An equation of a line can be graphed by substituting in at least three numbers for x to produce the corresponding values for y. • These are then the coordinates of three points on the line, which can be plotted and connected to form the line. An exponential function, y = ax, is a curved line that gets closer to but does not touch the x-axis. A line that comes close to but never touches the x-axis is called a horizontal asymptote. • An exponential function can be graphed by substituting numbers in for x and determining the value of y and then plotting the points on the coordinate plane. © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.
A quadratic function, y = ax² + bx +c, produces a parabola when graphed. A parabola is a U-shaped line that can either be facing up or facing down. • The point where a parabola is at its minimum or maximum is called the turning point or vertex. • The axis of symmetry is the line that cuts the parabola into two equal mirror images. • A quadratic function can be graphed by substituting numbers in for x and determining the value of y and then plotting the points on the coordinate plane. Functions represent variables and how they are related to one another. One type of relationship between variables is called inverse variation and means that as one variable increases, the other decreases. The formula for an inverse variation is xy = k, where k is the constant of variation. How to use functions A function is a rule that tells what needs to be done to an input to produce an output. If a function is given, simply substitute any given x values into the function to determine the y values. A linear function can always be put into the form, y = mx + b. To determine which function a graph represents, find the slope and y- intercept of the line. When the slope and y-intercept are substituted into the equation, the function is determined. The graph of a linear function is a straight line. An exponential function is always in the form of y = ax. When graphed it is a curved line that comes close to, but never touches the x-axis. A quadratic function is always in the form, y = ax² + bx +c. When graphed, it is a parabola. The axis of symmetry and turning point can be found with the equation, x = -b/2a, where a and b are the coefficients of the first and second terms in the equation. When x is found, it is substituted into the equation to find y, and therefore the coordinate of the turning point. © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.
Functions need to be recognized on a graph. The figures show the two types of nonlinear functions. Exponential function Quadratic function y = ax y = ax² + bx +c The y values for any function, whether linear or nonlinear, can be found by plugging in the x values and evaluating the function. Inverse variation is the relationship between two variables when one variable increases, the other variable decreases. The equation of inverse variation is xy = k, where k is the constant of variation. The table shows an inverse variation. © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.
Try This! 1. What are the y values for the linear function, y = -3x + 6 when the x values are -3, -2, -1? 2. What are the y values for the exponential function, y = 2x when the x values are -1, 0, 1? 3. What are the y values for the quadratic function, y = x² + 2x - 4 when the x values are 0, 1, 2? 4. Which type of function is -3x + 2y = -9? 5. If y varies inversely as x, and y = 18 when x = 2, what is y when x = 9? 6. If y varies inversely as x, and y = 7 when x = 5, what is the constant of variation? © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.
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