4. Describe the behavior of the graph at the x-intercepts for the function f(x) = (2x-7)^7 (x+3)^4. Be sure to identify each x-intercept and justify your answer. How does each exponent change the look of the graph at the x-axis?
Well, well, well, let's talk about these x-intercepts, shall we? The first thing we should do is find those little buggers. To do that, we set the function equal to zero and solve for x. In this case, we have f(x) = (2x-7)^7 (x+3)^4.
Now, let's find the x-intercepts. For the term (2x-7)^7, we set it equal to zero and solve for x. So, 2x-7 = 0, which means x = 7/2.
For the term (x+3)^4, we set it equal to zero and solve for x. That means x+3 = 0, which leads us to x = -3.
So, we have two x-intercepts: x = 7/2 and x = -3.
Now, let's see how those exponents are changing the look of the graph at the x-axis. The exponent of 7 for (2x-7)^7 means that the graph will touch and change direction at x = 7/2. It's like a doing a stunt on a motorcycle, flipping in mid-air.
The exponent of 4 for (x+3)^4 means that the graph will also touch and change direction at x = -3. It's like a gymnast doing a fancy somersault, gracefully landing on the x-axis.
Essentially, the higher the exponent, the more dramatic the change in the graph's behavior at the x-axis. It's like adding sprinkles to your ice cream. The more sprinkles you have, the more festive it becomes. So, those exponents are spicing things up at the x-intercepts, giving the graph some flair and excitement.
To describe the behavior of the graph at the x-intercepts of the function f(x) = (2x-7)^7 (x+3)^4, we first need to find the x-intercepts.
To find the x-intercepts, we set f(x) = 0 and solve for x:
(2x-7)^7 (x+3)^4 = 0
This equation will be satisfied if either (2x-7)^7 = 0 or (x+3)^4 = 0.
For (2x-7)^7 = 0, we solve for x:
2x - 7 = 0
2x = 7
x = 7/2
So the x-intercept at (7/2, 0) is due to the factor (2x-7)^7 = 0.
Now, for (x+3)^4 = 0, we solve for x:
x + 3 = 0
x = -3
So the x-intercept at (-3, 0) is due to the factor (x+3)^4 = 0.
The behavior of the graph at each x-intercept depends on the exponent of the corresponding factor.
For the x-intercept at (7/2, 0) resulting from (2x-7)^7 = 0, we have an odd exponent of 7.
When an odd exponent is applied to a factor, it means that the factor will change sign at that x-value. In this case, the value of (2x-7)^7 changes from positive to negative or from negative to positive at x = 7/2. As a result, the graph crosses the x-axis at this point.
For the x-intercept at (-3, 0) resulting from (x+3)^4 = 0, we have an even exponent of 4.
When an even exponent is applied to a factor, it means that the factor does not change sign at that x-value. In this case, the value of (x+3)^4 remains positive regardless of the value of x. As a result, the graph touches but does not cross the x-axis at this point.
In summary, the behavior of the graph at the x-intercept (7/2, 0) is a crossing point since the corresponding factor (2x-7)^7 changes sign, while the behavior at the x-intercept (-3, 0) is a touch point because the corresponding factor (x+3)^4 does not change sign.
To describe the behavior of the graph at the x-intercepts for the function f(x) = (2x-7)^7 (x+3)^4, we first need to find the x-intercepts and then analyze the behavior of the graph at those points.
Step 1: Finding the x-intercepts
The x-intercepts are the values of x for which f(x) equals zero. In other words, the x-intercepts are the values that make the function equal to zero. To find them, we set f(x) equal to zero and solve for x:
(2x-7)^7 (x+3)^4 = 0
When a product equals zero, at least one of the factors must be equal to zero. So, we can set each factor equal to zero and solve:
1) 2x-7 = 0
2x = 7
x = 7/2 = 3.5
2) x+3 = 0
x = -3
Therefore, the x-intercepts are located at x = 3.5 and x = -3.
Step 2: Analyzing the behavior of the graph at the x-intercepts
To analyze the behavior of the graph at each x-intercept, we can use the concept of the multiplicity of zeros. The multiplicity of a zero represents how many times a factor is repeated.
Let's consider each factor separately:
1) (2x-7)^7
The exponent 7 indicates that the factor has a multiplicity of 7. When a factor is repeated an odd number of times, the graph will cross the x-axis at that x-intercept. Since 7 is odd, the graph will cross the x-axis at x = 3.5.
2) (x+3)^4
The exponent 4 indicates that the factor has a multiplicity of 4. When a factor is repeated an even number of times, the graph will touch the x-axis but not cross it. Since 4 is even, the graph will touch the x-axis at x = -3.
Therefore, at x = 3.5, the graph will cross the x-axis, while at x = -3, the graph will touch the x-axis without crossing it.
In terms of how each exponent changes the look of the graph at the x-axis, a higher exponent increases the "flatness" or "steepness" of the graph near the x-axis. In this case, the factor (2x-7) with exponent 7 will create a steeper slope near the x-intercept, while the factor (x+3) with exponent 4 will produce a flatter slope near the x-intercept.
roots with odd multiplicity: graph crosses the x-axis
with even multiplicity, the graph bounces off the axis.
compare x^2 and x^3