Question
You will need two sheets of paper for this part of the portfolio: One paper for answering and showing your work on questions 1, 2, and 5 and then one paper for the hand-drawn graph. Upload pictures of both sheets of paper to the dropbox.
1. Find the zeros of the function h(x)=−13x2(5x+10)(x−3)
. Show your handwritten work. Write your zeros as ordered pairs like (x, y).
2. Pick two x-values that are between the zeros you found in the first question. Plug those x-values into the function to find the corresponding y-values. Show your handwritten work. Write your answers as ordered pairs like (x, y).
3. Make a hand-drawn graph of each point you found in parts 1 and 2. You should have three zeros as well as two points between those zeros for a total of 5 points on your graph.
4. Finish your graph. Scale your y-axis so you can see all relative extrema. Make sure to label all 5 points, label your x and y axes, and put arrows on both ends of your graph.
5. Describe the end behavior of h(x), explain completely. Make sure to write about how the degree and leading coefficient affect end behavior.
1. Find the zeros of the function h(x)=−13x2(5x+10)(x−3)
. Show your handwritten work. Write your zeros as ordered pairs like (x, y).
2. Pick two x-values that are between the zeros you found in the first question. Plug those x-values into the function to find the corresponding y-values. Show your handwritten work. Write your answers as ordered pairs like (x, y).
3. Make a hand-drawn graph of each point you found in parts 1 and 2. You should have three zeros as well as two points between those zeros for a total of 5 points on your graph.
4. Finish your graph. Scale your y-axis so you can see all relative extrema. Make sure to label all 5 points, label your x and y axes, and put arrows on both ends of your graph.
5. Describe the end behavior of h(x), explain completely. Make sure to write about how the degree and leading coefficient affect end behavior.
Answers
GPT-4o mini
To complete this portfolio assignment, let's go through the steps requested for each question.
### 1. Finding the Zeros of \( h(x) = -13x^2(5x + 10)(x - 3) \)
To find the zeros, we will set \( h(x) = 0 \):
\[ -13x^2(5x + 10)(x - 3) = 0 \]
This equation holds true if any of its factors are zero:
1. **First Factor**: \( -13x^2 = 0 \)
- \( x^2 = 0 \)
- \( x = 0 \)
2. **Second Factor**: \( 5x + 10 = 0 \)
- \( 5x = -10 \)
- \( x = -2 \)
3. **Third Factor**: \( x - 3 = 0 \)
- \( x = 3 \)
Thus, the zeros of the function are \( x = 0, -2, \) and \( 3 \). We can express these as ordered pairs with \( y = 0 \):
- \( (0, 0) \)
- \( (-2, 0) \)
- \( (3, 0) \)
### 2. Picking Two X-Values Between Zeros
The zeros we found are \( -2, 0, \) and \( 3 \). We will choose two \( x \)-values between these zeros: \( -1 \) and \( 1 \).
#### For \( x = -1 \):
\[ h(-1) = -13(-1)^2(5(-1) + 10)((-1) - 3) \]
Calculating each part:
- \( (-1)^2 = 1 \)
- \( 5(-1) + 10 = -5 + 10 = 5 \)
- \( (-1) - 3 = -4 \)
Calculating \( h(-1) \):
\[ h(-1) = -13(1)(5)(-4) = -13 \cdot 5 \cdot -4 = 260 \]
So, the ordered pair is:
- \( (-1, 260) \)
#### For \( x = 1 \):
\[ h(1) = -13(1)^2(5(1) + 10)(1 - 3) \]
Calculating each part:
- \( (1)^2 = 1 \)
- \( 5(1) + 10 = 5 + 10 = 15 \)
- \( (1) - 3 = -2 \)
Calculating \( h(1) \):
\[ h(1) = -13(1)(15)(-2) = -13 \cdot 15 \cdot -2 = 390 \]
So, the ordered pair is:
- \( (1, 390) \)
### Summary of Coordinates
- **Zeros**:
- \( (0, 0) \)
- \( (-2, 0) \)
- \( (3, 0) \)
- **Points between zeros**:
- \( (-1, 260) \)
- \( (1, 390) \)
### 3. Hand-Drawn Graph
On the graphing paper, you will:
- Mark the zeros \( (0, 0) \), \( (-2, 0) \), \( (3, 0) \).
- Plot the points \( (-1, 260) \) and \( (1, 390) \).
- Label each point accordingly.
- Scale axes appropriately and use arrows indicating the ends of the graph.
### 4. Complete the Graph with Axes and Points Labeled
- Here you will ensure that both axes are marked with appropriate scales to show all behaviors of the graph.
### 5. Describe the End Behavior of \( h(x) \)
The function \( h(x) = -13x^2(5x + 10)(x - 3) \) is a polynomial of degree 4 (2 from \( x^2 \) and 1 from each factor).
- The leading coefficient is negative (-13).
- For polynomial functions, the end behavior is determined by the degree and leading coefficient:
- A polynomial of even degree with a negative leading coefficient will have both ends of the graph going downwards.
Thus, as \( x \to -\infty, h(x) \to -\infty \) and as \( x \to \infty, h(x) \to -\infty \).
In summary, the graph starts in the upper left quadrant and falls to the lower right quadrant.
### Final Steps
- Once you have performed the calculations and sketches, take clear photographs of both your work and drawing, then upload them to Dropbox as instructed.
### 1. Finding the Zeros of \( h(x) = -13x^2(5x + 10)(x - 3) \)
To find the zeros, we will set \( h(x) = 0 \):
\[ -13x^2(5x + 10)(x - 3) = 0 \]
This equation holds true if any of its factors are zero:
1. **First Factor**: \( -13x^2 = 0 \)
- \( x^2 = 0 \)
- \( x = 0 \)
2. **Second Factor**: \( 5x + 10 = 0 \)
- \( 5x = -10 \)
- \( x = -2 \)
3. **Third Factor**: \( x - 3 = 0 \)
- \( x = 3 \)
Thus, the zeros of the function are \( x = 0, -2, \) and \( 3 \). We can express these as ordered pairs with \( y = 0 \):
- \( (0, 0) \)
- \( (-2, 0) \)
- \( (3, 0) \)
### 2. Picking Two X-Values Between Zeros
The zeros we found are \( -2, 0, \) and \( 3 \). We will choose two \( x \)-values between these zeros: \( -1 \) and \( 1 \).
#### For \( x = -1 \):
\[ h(-1) = -13(-1)^2(5(-1) + 10)((-1) - 3) \]
Calculating each part:
- \( (-1)^2 = 1 \)
- \( 5(-1) + 10 = -5 + 10 = 5 \)
- \( (-1) - 3 = -4 \)
Calculating \( h(-1) \):
\[ h(-1) = -13(1)(5)(-4) = -13 \cdot 5 \cdot -4 = 260 \]
So, the ordered pair is:
- \( (-1, 260) \)
#### For \( x = 1 \):
\[ h(1) = -13(1)^2(5(1) + 10)(1 - 3) \]
Calculating each part:
- \( (1)^2 = 1 \)
- \( 5(1) + 10 = 5 + 10 = 15 \)
- \( (1) - 3 = -2 \)
Calculating \( h(1) \):
\[ h(1) = -13(1)(15)(-2) = -13 \cdot 15 \cdot -2 = 390 \]
So, the ordered pair is:
- \( (1, 390) \)
### Summary of Coordinates
- **Zeros**:
- \( (0, 0) \)
- \( (-2, 0) \)
- \( (3, 0) \)
- **Points between zeros**:
- \( (-1, 260) \)
- \( (1, 390) \)
### 3. Hand-Drawn Graph
On the graphing paper, you will:
- Mark the zeros \( (0, 0) \), \( (-2, 0) \), \( (3, 0) \).
- Plot the points \( (-1, 260) \) and \( (1, 390) \).
- Label each point accordingly.
- Scale axes appropriately and use arrows indicating the ends of the graph.
### 4. Complete the Graph with Axes and Points Labeled
- Here you will ensure that both axes are marked with appropriate scales to show all behaviors of the graph.
### 5. Describe the End Behavior of \( h(x) \)
The function \( h(x) = -13x^2(5x + 10)(x - 3) \) is a polynomial of degree 4 (2 from \( x^2 \) and 1 from each factor).
- The leading coefficient is negative (-13).
- For polynomial functions, the end behavior is determined by the degree and leading coefficient:
- A polynomial of even degree with a negative leading coefficient will have both ends of the graph going downwards.
Thus, as \( x \to -\infty, h(x) \to -\infty \) and as \( x \to \infty, h(x) \to -\infty \).
In summary, the graph starts in the upper left quadrant and falls to the lower right quadrant.
### Final Steps
- Once you have performed the calculations and sketches, take clear photographs of both your work and drawing, then upload them to Dropbox as instructed.