https://file.aiquickdraw.com/mathbot/file/1728664080077xeaxxc81.jpg

Let's break down the questions from the image:

### **Patterns Section #3: Sequence**
The equation for the sequence is given by:
\[
t_n = 5 - 3(n - 1)
\]
We will need to:
1. **a. Determine the 5th value of the sequence** and plot it on the graph.
2. **b. Determine the 6th value of the sequence** and plot it on the graph.
3. **c. Explain how to determine the initial value of the sequence.**

### **Functions 1 Section #1: Quadratic Functions**
You need to choose one of the quadratic functions to analyze:
1. \( y = -x^2 + 9x - 14 \)
2. \( y = x^2 + 14x + 40 \)

For the selected quadratic function, you need to fill in:
1. **Orientation (Concavity)**: Whether the parabola opens upwards or downwards.
2. **Minimum or Maximum**: Determine if the vertex is a minimum or maximum.
3. **X-Intercepts**: Solve for the x-intercepts (if they exist).
4. **Y-Intercept**: Find the y-intercept.
5. **Vertex**: Calculate the vertex of the parabola.

---

Let's start by solving the sequence problem.

#### Part (a) & (b): Values of the Sequence

For the sequence:
\[
t_n = 5 - 3(n - 1)
\]
We'll calculate the values for \(n = 5\) and \(n = 6\).

For \(n = 5\):
\[
t_5 = 5 - 3(5 - 1) = 5 - 3(4) = 5 - 12 = -7
\]

For \(n = 6\):
\[
t_6 = 5 - 3(6 - 1) = 5 - 3(5) = 5 - 15 = -10
\]

So, the 5th value of the sequence is \(-7\) and the 6th value is \(-10\).

#### Part (c): Initial Value of the Sequence
The initial value is \( t_1 \). We can determine it by plugging \( n = 1 \) into the formula:
\[
t_1 = 5 - 3(1 - 1) = 5 - 3(0) = 5
\]
Thus, the initial value is \( t_1 = 5 \).

---

Would you like help with the quadratic function analysis next? Let me know if you'd like more details or explanations on any part.

### Relative Questions:
1. How do you find the general formula for an arithmetic sequence?
2. What does the slope of a sequence tell you about its rate of change?
3. How can you find the x-intercepts of a quadratic function?
4. What does the vertex of a quadratic function represent?
5. How do you determine if a quadratic function has a maximum or minimum value?

### Tip:
For quadratic functions, completing the square can help you find the vertex quickly if you're working with a complicated equation.

Use the information provided to solve a sequence and demonstrate knowledge of quadratic functions in standard form.

Learn how to solve an arithmetic sequence and analyze quadratic functions in standard form. This exercise covers key algebraic concepts such as concavity, intercepts, and finding the vertex of a parabola. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation