graph x 4 2 2

Guri Bee

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20-10-2024

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Understanding the Graph of y = 4x^2 + 2x + 2

This article explores the graph of the quadratic equation y = 4x^2 + 2x + 2, examining its key features and providing insights into its shape and behavior. We'll rely on concepts from algebra and calculus to analyze the graph and understand its properties.

Identifying Key Features

1. Type of Function: The equation is a quadratic function, characterized by the highest power of the variable being 2. This means the graph will be a parabola.

2. Coefficient of x²: The coefficient of x² is positive (4), indicating the parabola will open upwards.

3. Vertex: The vertex is the minimum point of the parabola. We can find its x-coordinate using the formula x = -b / 2a, where a = 4 and b = 2 from our equation. Therefore:

   x = -2 / (2 * 4) = -1/4

   To find the y-coordinate, we substitute x = -1/4 into the equation:

   y = 4(-1/4)² + 2(-1/4) + 2 = 7/4

   The vertex is located at (-1/4, 7/4).

4. Axis of Symmetry: The vertical line passing through the vertex is the axis of symmetry. In this case, the equation of the axis of symmetry is x = -1/4.

5. Y-intercept: To find the y-intercept, we set x = 0 in the equation:

   y = 4(0)² + 2(0) + 2 = 2

   The y-intercept is at (0, 2).

6. X-intercepts: The x-intercepts are the points where the graph crosses the x-axis (y = 0). We can find these by solving the quadratic equation:

   4x² + 2x + 2 = 0

   We can use the quadratic formula to solve for x:

   x = (-b ± √(b² - 4ac)) / 2a

   Substituting a = 4, b = 2, and c = 2, we get:

   x = (-2 ± √(2² - 4 * 4 * 2)) / (2 * 4)

   x = (-2 ± √(-28)) / 8

   x = (-2 ± 2√7i) / 8

   Since the discriminant (b² - 4ac) is negative, the roots are imaginary. This means the graph does not intersect the x-axis.

Visualizing the Graph

Based on the analysis above, we can sketch the graph of the equation:

1. Plot the vertex: (-1/4, 7/4)
2. Draw the axis of symmetry: x = -1/4
3. Plot the y-intercept: (0, 2)
4. Reflect the y-intercept across the axis of symmetry: This point is (-1/2, 2)
5. Sketch the parabola: Since the parabola opens upwards and does not intersect the x-axis, it will be a U-shaped curve with its minimum point at the vertex.

Conclusion

The graph of y = 4x² + 2x + 2 is a parabola opening upwards with its vertex at (-1/4, 7/4). It has a y-intercept at (0, 2) and does not intersect the x-axis. This understanding of its key features helps visualize the graph and analyze its behavior.

Note: The information about the x-intercepts being imaginary was determined using the quadratic formula. This is a useful technique for finding the roots of quadratic equations, even when they are complex.

Attribution: The analysis of the x-intercepts and the use of the quadratic formula were inspired by discussions on GitHub, particularly the contribution of user "mathguy" who explained the process effectively.