Graphing+Quadratic+Functions-Target+F-Comparing+Functions-Guided+Learning

Target F: Compare linear, exponential, and quadratic functions from any representation.
Recall that the graph of a linear function is a line (the work line is in the word linear!). The graph of an exponential function looks like a curve that decreases or increases rapidly, and the graph of a quadratic function is a parabola with either a high point or low point. Take a look at the graphs below to see a visual of what each type of function looks like.
 * Comparing Linear, Exponential, and Quadratic Functions by Analyzing Graphs **

Each type of function has its own characteristic to help distinguish between whether a function is linear, exponential, or quadratic. A linear function contains a polynomial expression where the highest degree is one, and is typically written in slope-intercept form. An exponential function is written in the form: f(x) = ab x, containing a variable exponent. A quadratic function has a degree of 2, and can also be written in three different forms: standard, vertex, and factored form (although there is not a visible exponent of 2 when a quadratic function is written in factored form, when the factors are multiplied out, they will generate a function with a degree of 2). The key characteristic is the highest exponent in the function has a value of 2. Here are examples of each type of function:
 * Comparing Linear, Exponential, and Quadratic Functions by Analyzing Equations**
 * Linear Function: f(x) = 3x - 2
 * Exponential Function: f(x) = 4(3) x
 * Quadratic Function: f(x) = 2x 2 + 10x + 4; f(x) = (x - 5) 2 + 6; f(x) = (x - 5)(x + 3)

In order to determine if a table of values represents a linear, exponential, or quadratic function, you need to look at the differences between the x-values and y-values in the table. If the x-values are increasing by an equal increment of 1, and the difference in the y-values is the same, the table represents a linear function. In a table representing an exponential function, if the x-values increase by an equal increment of 1, then the y-values will change by a common ratio. A table that represents a quadratic function is slightly more complex. In order to determine if a table represents a quadratic function, you need to find the second differences for the table. To do this, find the difference between each of the y-values. and then find the difference of those values (the second differences). If the value of the second differences is all the same (and the x-values all increase by 1), then the table represents a quadratic function. Look at the examples below:
 * Comparing Linear, Exponential, and Quadratic Functions by Analyzing Tables**

When comparing functions by evaluating, substitute in the given value for the variable into each function and evaluate. For example, given the functions h(x) = 6x - 5 and k(x) = 2x 2 -3x + 1, determine which has the greater value: h(4) or k(-3). To find h(4), substitute 4 in for the variable in the function h(x) = 6x - 5. To find k(-3), substitute -3 in for the variable in the function k(x) = 2x 2 - 3x + 1. Since h(4) = 19 and k(-3) = 26, k(-3) has the greater value.
 * Comparing Functions by Evaluating**

As with evaluating, you can compare different types of functions and their values by looking at their graphs. Use the graphs below to determine if j(-1) is greater than, less than, or equal to g(4).
 * Comparing Functions by Graphing**

For the graph of j(x), if we want to know what the value of j(-1), move along the x-axis until you reach x = -1, then travel either up or down in order to hit the graph of the function. For j(-1), move left to where x is -1, and then if you travel down, you will hit the graph where y is -4. So j(-1) = -4. For the graph of g(x), we want to know what the value of g(4) is. To find this, move along the x-axis until you reach x = 4, then you have to travel up in order to hit the graph of function, which is where y is 1. So g(4) = 1. Since j(-1) =-4 and g(4) = 1, j(-1) < g(4), because -4 is less than 1.

Use the function, table, and graph of the following functions to answer the questions below.
 * Graphing Quadratic Functions-Target F-Comparing Functions-Quick Check**

1) Which type of function does the table of j(x) represent?

2) Determine if g(2) is greater than, less than, or equal to h(4).

Graphing Quadratic Functions-Target F-Comparing Functions-Quick Check Solutions