For the following exercises, draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima, inflection points, and asymptotic behavior.

294.

y=3x2+2x+4

Let's analyze the function \( y = 3x^2 + 2x + 4 \) and graph it without using a calculator. 

### 1. **Identify the Type of Function:**
   - The function \( y = 3x^2 + 2x + 4 \) is a quadratic function, and its graph will be a parabola. Since the coefficient of \( x^2 \) is positive (3), the parabola opens upwards.

### 2. **Find the Vertex:**
   - The vertex form of a quadratic function is \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola.
   - To find the vertex for \( y = 3x^2 + 2x + 4 \), we use the vertex formula:
     \[
     h = -\frac{b}{2a} = -\frac{2}{2 \times 3} = -\frac{2}{6} = -\frac{1}{3}
     \]
   - Substitute \( h = -\frac{1}{3} \) back into the equation to find \( k \):
     \[
     k = 3\left(-\frac{1}{3}\right)^2 + 2\left(-\frac{1}{3}\right) + 4
     \]
     \[
     k = 3\left(\frac{1}{9}\right) - \frac{2}{3} + 4 = \frac{1}{3} - \frac{2}{3} + 4 = \frac{-1}{3} + 4 = \frac{11}{3}
     \]
   - The vertex is \(\left(-\frac{1}{3}, \frac{11}{3}\right)\).

### 3. **Determine the Axis of Symmetry:**
   - The axis of symmetry is the vertical line through the vertex:
     \[
     x = -\frac{1}{3}
     \]

### 4. **Find the Y-Intercept:**
   - The y-intercept occurs when \( x = 0 \):
     \[
     y = 3(0)^2 + 2(0) + 4 = 4
     \]
   - The y-intercept is \( (0, 4) \).

### 5. **Find the X-Intercepts:**
   - The x-intercepts occur where \( y = 0 \):
     \[
     3x^2 + 2x + 4 = 0
     \]
   - Solving this quadratic equation using the discriminant:
     \[
     \text{Discriminant} = b^2 - 4ac = 2^2 - 4(3)(4) = 4 - 48 = -44
     \]
   - Since the discriminant is negative, there are no real x-intercepts. The parabola does not cross the x-axis.

### 6. **Determine the Asymptotic Behavior:**
   - Since this is a polynomial function (quadratic), it has no asymptotes. As \( x \) approaches infinity or negative infinity, the function will continue to increase due to the positive leading coefficient.

### 7. **Sketch the Graph:**
   - Plot the vertex \(\left(-\frac{1}{3}, \frac{11}{3}\right)\).
   - Draw the axis of symmetry \( x = -\frac{1}{3} \).
   - Plot the y-intercept \( (0, 4) \).
   - Since the parabola opens upwards and does not have real x-intercepts, sketch the parabola passing through these points with the correct curvature.

### 8. **Summary of Features:**
   - **Vertex:** \( \left(-\frac{1}{3}, \frac{11}{3}\right) \)
   - **Axis of Symmetry:** \( x = -\frac{1}{3} \)
   - **Y-Intercept:** \( (0, 4) \)
   - **X-Intercepts:** None (no real solutions)
   - **Opening:** Upward
   - **Asymptotes:** None

Would you like a more detailed explanation of any part of this process or have any questions?

---

Here are some related questions you might consider:

1. How would the graph change if the coefficient of \( x^2 \) were negative?
2. What is the effect of changing the linear coefficient \( b \) on the vertex of the parabola?
3. How do you determine the direction of opening for a quadratic function?
4. What is the significance of the discriminant in finding the x-intercepts?
5. How would you graph \( y = 3x^2 + 2x + 4 \) using transformations from the parent function \( y = x^2 \)?
6. How can you convert the given quadratic equation into vertex form?
7. What is the role of the constant term in a quadratic function?
8. How do you determine the concavity of a parabola from the quadratic function?

**Tip:** The discriminant in a quadratic equation provides key information about the nature of the roots—whether they are real or complex, and if real, whether they are distinct or repeated.

Learn how to graph the quadratic function y = 3x^2 + 2x + 4 without using a calculator. Understand the vertex, axis of symmetry, y-intercept, and asymptotic behavior of the graph. Suitable for high school students studying quadratic functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation