Choose The Graph Of Y X2 4x 5

Choosing the Graph of y = x² + 4x + 5: A Comprehensive Guide

Understanding quadratic functions and their graphical representations is crucial in algebra and beyond. This article will guide you through the process of identifying the correct graph for the quadratic equation y = x² + 4x + 5, covering key concepts and techniques. We'll explore multiple approaches, ensuring a thorough understanding of how to analyze and visualize quadratic functions.

Understanding Quadratic Functions

A quadratic function is a polynomial function of degree two, generally expressed in the form:

y = ax² + bx + c

where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The graph of a quadratic function is always a parabola, a symmetrical U-shaped curve. The characteristics of this parabola are determined by the values of 'a', 'b', and 'c'.

Key Features of a Parabola

Several key features help us identify and sketch parabolas:

• Vertex: The vertex is the lowest (minimum) or highest (maximum) point on the parabola. Its x-coordinate is given by x = -b / 2a. The y-coordinate is found by substituting this x-value back into the quadratic equation.

• Axis of Symmetry: This is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. Its equation is x = -b / 2a.

• x-intercepts (Roots or Zeros): These are the points where the parabola intersects the x-axis (where y = 0). They are found by solving the quadratic equation ax² + bx + c = 0. This can be done using factoring, the quadratic formula, or completing the square.

• y-intercept: This is the point where the parabola intersects the y-axis (where x = 0). It is simply the value of 'c' in the equation, i.e., (0, c).

• Concavity: The parabola opens upwards (concave up) if 'a' > 0, and opens downwards (concave down) if 'a' < 0.

Analyzing y = x² + 4x + 5

Now, let's apply these concepts to the specific quadratic function:

y = x² + 4x + 5

In this equation, a = 1, b = 4, and c = 5.

1. Determining the Vertex

The x-coordinate of the vertex is:

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

Substituting x = -2 back into the equation:

y = (-2)² + 4(-2) + 5 = 4 - 8 + 5 = 1

Therefore, the vertex is (-2, 1).

2. Identifying the Axis of Symmetry

The axis of symmetry is a vertical line passing through the vertex. Its equation is:

x = -2

3. Finding the x-intercepts

To find the x-intercepts, we set y = 0 and solve for x:

x² + 4x + 5 = 0

We can use the quadratic formula to solve this equation:

x = [-b ± √(b² - 4ac)] / 2a

x = [-4 ± √(4² - 4 * 1 * 5)] / (2 * 1)

x = [-4 ± √(-4)] / 2

Since the discriminant (b² - 4ac = -4) is negative, there are no real x-intercepts. This means the parabola does not intersect the x-axis.

4. Determining the y-intercept

The y-intercept is the value of y when x = 0:

y = (0)² + 4(0) + 5 = 5

Therefore, the y-intercept is (0, 5).

5. Determining Concavity

Since a = 1 (which is positive), the parabola opens upwards (concave up).

Sketching the Parabola

Based on our analysis:

• The vertex is at (-2, 1).
• The axis of symmetry is x = -2.
• There are no x-intercepts.
• The y-intercept is at (0, 5).
• The parabola opens upwards.

Using this information, we can accurately sketch the parabola. It will be a U-shaped curve opening upwards, with its lowest point at (-2, 1), passing through (0, 5), and never intersecting the x-axis.

Choosing the Correct Graph

When presented with multiple graph options, look for the one that matches all the characteristics we've determined:

• Vertex at (-2, 1): The lowest point of the parabola should be at this coordinate.
• Axis of symmetry at x = -2: The parabola should be symmetrical about this vertical line.
• No x-intercepts: The parabola should not cross the x-axis.
• y-intercept at (0, 5): The parabola should pass through this point.
• Opens upwards: The parabola should be U-shaped, not an inverted U.

Any graph that doesn't meet all these criteria is incorrect.

Alternative Approaches

While the above method is comprehensive, other approaches can help verify your choice:

Completing the Square

Completing the square transforms the quadratic equation into vertex form:

y = a(x - h)² + k

where (h, k) is the vertex. Let's apply this to our equation:

y = x² + 4x + 5

y = (x² + 4x + 4) + 1 (We add and subtract 4 to complete the square)

y = (x + 2)² + 1

This confirms our vertex at (-2, 1).

Using a Graphing Calculator or Software

Graphing calculators or software like Desmos or GeoGebra can quickly plot the quadratic function, visually confirming the parabola's shape and key features. This is a useful tool for verification and exploration.

Conclusion

Choosing the correct graph for y = x² + 4x + 5 involves a systematic analysis of the quadratic function's characteristics. By determining the vertex, axis of symmetry, x-intercepts (or lack thereof), y-intercept, and concavity, you can accurately identify the parabola's shape and position. Remember to check all these features against any given graph options to ensure a correct selection. Using supplementary methods like completing the square or graphing software can provide additional confirmation and enhance your understanding of quadratic functions. Mastering these techniques is fundamental to success in algebra and related fields.