Graphing F(x) = X² + 6x + 5 A Step-by-Step Guide

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Determining the graph of a quadratic function like f(x) = x² + 6x + 5 involves understanding the key features of parabolas. This article will guide you through the process of identifying the correct graph by analyzing the function's properties, including its vertex, axis of symmetry, intercepts, and concavity. We will explore each of these elements in detail, providing you with a comprehensive approach to graphing quadratic functions. The ultimate goal is to equip you with the knowledge and skills to confidently identify the graph of any quadratic function.

Understanding Quadratic Functions

Quadratic functions are polynomial functions of the second degree, generally expressed in the form f(x) = ax² + bx + c, where a, b, and c are constants, and a is not equal to zero. The graph of a quadratic function is a parabola, a U-shaped curve that opens either upwards or downwards. The shape and position of the parabola are determined by the coefficients a, b, and c. Understanding these coefficients is crucial for sketching or identifying the graph of a quadratic function. For instance, the sign of a dictates the parabola's concavity: if a is positive, the parabola opens upwards, and if a is negative, it opens downwards. The vertex, the turning point of the parabola, is another critical feature. Its coordinates can be calculated using the formula (-b/2a, f(-b/2a)). The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. The x-intercepts (if any) are the points where the parabola intersects the x-axis, and they can be found by setting f(x) to zero and solving for x. The y-intercept is the point where the parabola intersects the y-axis, and it can be found by setting x to zero. By analyzing these elements, we can accurately sketch or identify the graph of a quadratic function.

Key Features of the Function f(x) = x² + 6x + 5

To accurately graph or identify the graph of f(x) = x² + 6x + 5, we need to pinpoint its key features. Let's start by identifying the coefficients: a = 1, b = 6, and c = 5. Since a is positive (1 > 0), the parabola opens upwards. This means the parabola will have a minimum point, which is its vertex. Now, let's find the vertex. The x-coordinate of the vertex is given by -b/2a, which in this case is -6 / (2 * 1) = -3. To find the y-coordinate, we substitute x = -3 into the function: f(-3) = (-3)² + 6(-3) + 5 = 9 - 18 + 5 = -4. Therefore, the vertex of the parabola is at the point (-3, -4). The axis of symmetry is a vertical line that passes through the vertex, so its equation is x = -3. Next, let's find the x-intercepts by setting f(x) to zero and solving for x: x² + 6x + 5 = 0. This quadratic equation can be factored as (x + 1)(x + 5) = 0, which gives us two solutions: x = -1 and x = -5. These are the x-intercepts of the parabola. To find the y-intercept, we set x = 0: f(0) = 0² + 6(0) + 5 = 5. So, the y-intercept is at the point (0, 5). By analyzing these key features – the upward-opening parabola, the vertex at (-3, -4), the axis of symmetry at x = -3, the x-intercepts at x = -1 and x = -5, and the y-intercept at (0, 5) – we can accurately sketch or identify the graph of the function f(x) = x² + 6x + 5.

Determining the Correct Graph

With the key features of f(x) = x² + 6x + 5 identified, we can now focus on determining the correct graph. We know the parabola opens upwards, has a vertex at (-3, -4), intersects the x-axis at -1 and -5, and the y-axis at 5. When presented with multiple graph options, we can systematically eliminate those that do not match these characteristics. First, any graph that opens downwards can be immediately ruled out, as our function has a positive coefficient for the term. Next, we look for the vertex. The correct graph must have its lowest point (the vertex) at the coordinates (-3, -4). Graphs with vertices in other locations can be eliminated. The axis of symmetry, x = -3, should also be visually verifiable; the parabola should be symmetrical about this vertical line. The x-intercepts are crucial points. The graph should cross the x-axis at x = -1 and x = -5. If a graph shows different x-intercepts, it is not the correct representation of the function. Lastly, the y-intercept should be at (0, 5). The graph should intersect the y-axis at this point. By carefully checking each of these features against the provided graph options, we can confidently identify the one that accurately represents the quadratic function f(x) = x² + 6x + 5. This methodical approach ensures that we select the correct graph based on the function's inherent properties.

Step-by-Step Graphing of f(x) = x² + 6x + 5

To further solidify our understanding, let's walk through the step-by-step process of graphing f(x) = x² + 6x + 5. This process not only aids in identifying the correct graph among options but also builds a strong foundation for graphing any quadratic function.

  1. Determine the direction of the parabola: As we established earlier, since a = 1 (positive), the parabola opens upwards.
  2. Find the vertex: The x-coordinate of the vertex is -b/2a = -6 / (2 * 1) = -3. The y-coordinate is f(-3) = (-3)² + 6(-3) + 5 = -4. So, the vertex is at (-3, -4).
  3. Draw the axis of symmetry: This is a vertical line passing through the vertex, so its equation is x = -3. Draw a dashed line on the graph to represent this axis.
  4. Find the x-intercepts: Set f(x) = 0 and solve for x: x² + 6x + 5 = 0. Factoring gives us (x + 1)(x + 5) = 0, so the x-intercepts are x = -1 and x = -5. Plot these points on the graph.
  5. Find the y-intercept: Set x = 0: f(0) = 0² + 6(0) + 5 = 5. Plot the y-intercept at (0, 5).
  6. Plot additional points (optional): To get a more accurate graph, you can choose additional x-values and calculate their corresponding y-values. For example, you could calculate f(-2) and f(-4), which will be symmetrical around the axis of symmetry.
  7. Sketch the parabola: Draw a smooth, U-shaped curve that passes through the intercepts and the vertex. The parabola should be symmetrical about the axis of symmetry and open upwards. This step-by-step approach ensures a precise and accurate graph of the quadratic function.

Common Mistakes to Avoid

When working with quadratic functions and their graphs, several common mistakes can lead to incorrect results. Being aware of these pitfalls can significantly improve accuracy. One frequent error is miscalculating the vertex. The formula -b/2a must be applied carefully, and the resulting x-coordinate must be correctly substituted back into the function to find the y-coordinate. A sign error in this calculation can shift the entire parabola. Another common mistake is incorrectly factoring the quadratic equation to find the x-intercepts. If factoring is challenging, the quadratic formula can be used as an alternative, but it must be applied with precision. Overlooking the sign of the leading coefficient a is another error. This coefficient determines whether the parabola opens upwards or downwards, and a mistake here will result in a graph with the wrong orientation. Forgetting to plot the y-intercept can also lead to an incomplete or inaccurate graph. The y-intercept provides a crucial point for sketching the curve. Finally, sketching a parabola that is not symmetrical about its axis of symmetry is a common visual error. The parabola should be a perfectly symmetrical U-shape, and any asymmetry indicates a mistake in plotting or sketching. By avoiding these common pitfalls, you can ensure that your graphs of quadratic functions are accurate and reliable.

Conclusion

In conclusion, identifying the graph of a quadratic function like f(x) = x² + 6x + 5 involves a systematic analysis of its key features. By determining the direction of the parabola (upwards or downwards), finding the vertex, axis of symmetry, x-intercepts, and y-intercept, we can accurately sketch or select the correct graph. The step-by-step graphing process provides a structured approach to plotting the parabola, ensuring precision and accuracy. Avoiding common mistakes such as miscalculating the vertex or incorrectly factoring the quadratic equation is crucial for success. A thorough understanding of these concepts and techniques empowers you to confidently graph and analyze quadratic functions. The ability to translate a quadratic equation into its graphical representation is a fundamental skill in mathematics, with applications spanning various fields, including physics, engineering, and economics. By mastering this skill, you gain a valuable tool for problem-solving and analytical thinking. Therefore, a comprehensive grasp of graphing quadratic functions is essential for any student or professional seeking to excel in mathematical endeavors.