Suppose a quadratic function \( f(x) \) can be factored into the product of two linear factors \( p(x) \) and \( q(x) \), so that we can write \( f(x)=p(x) q(x) \). How do the x-intercepts of the linear functions \( y= p(x) \) and \( y= q(x) \) relate to the x-intercepts of the quadratic function \( f(x) \) ?
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Comparing Linear, Exponential, and Quadratic Functions by Analyzing Graphs
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.
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 Equations
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: \( f(x) = a(b)^x \), 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 Tables
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, remember this for when you take calculus 1 and it will all make sense! Look at the examples below:
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: \( f(x) = a(b)^x \), 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:
- Linear Function: \( f(x) = 3x - 2 \)
- Exponential Function: \( f(x) = 4(3)^x \)
- Quadratic Function: \( f(x) = 2x^2 + 10x + 4; f(x) = (x - 5)^2 + 6; f(x) = (x - 5)(x + 3) \)
Comparing Linear, Exponential, and Quadratic Functions by Analyzing Tables
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, remember this for when you take calculus 1 and it will all make sense! Look at the examples below:
Comparing Functions by Evaluating
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 \( k(x) = 2x^2 - 3x + 1 \).
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 Graphing
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
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 \( g(4) \).
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.
Quick Check
Use the function, table, and graph of the following functions to answer the questions below.
Use the function, table, and graph of the following functions to answer the questions below.
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 \( h(4) \).
Quick Check Solutions
2) Determine if \( g(2) \) is greater than, less than, or equal to \( h(4) \).
Quick Check Solutions