X-Intercept And Vertex Of G(x) = -5(x-3)^2 A Comprehensive Guide

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Understanding quadratic functions is a fundamental concept in mathematics, with applications spanning various fields from physics to economics. These functions, characterized by their parabolic curves, hold a wealth of information within their equations. In this comprehensive guide, we will delve into the specifics of the quadratic function g(x) = -5(x-3)^2, meticulously dissecting it to pinpoint its x-intercept and vertex. These two features are pivotal in grasping the behavior and graphical representation of any quadratic function.

The x-intercept, the point where the parabola intersects the x-axis, reveals the real roots of the quadratic equation. Conversely, the vertex signifies the parabola's extreme point – either its maximum or minimum value. By mastering the techniques to identify these key features, you'll unlock a deeper understanding of quadratic functions and their significance.

Decoding the Quadratic Equation: A Journey into g(x) = -5(x-3)^2

The given quadratic function, g(x) = -5(x-3)^2, is presented in vertex form. This form, expressed as g(x) = a(x-h)^2 + k, provides immediate insights into the parabola's vertex and its orientation. In this equation, (h, k) represents the vertex, and 'a' dictates the parabola's direction and width. A negative 'a' value, as in our case (-5), indicates that the parabola opens downwards, implying a maximum value at the vertex. The magnitude of 'a' signifies the steepness of the parabola; a larger absolute value denotes a narrower parabola.

To find the x-intercept, we set g(x) equal to zero and solve for x. This is because the x-intercept occurs where the function's output (g(x)) is zero, meaning the parabola intersects the x-axis at that point. Substituting g(x) with 0 in our equation gives us 0 = -5(x-3)^2. Dividing both sides by -5 simplifies the equation to 0 = (x-3)^2. Taking the square root of both sides yields 0 = x-3, which then solves to x = 3. This indicates that the parabola intersects the x-axis at x = 3. Since the x-intercept is a point on the coordinate plane, we express it as an ordered pair: (3, 0).

The vertex of the parabola is directly discernible from the vertex form of the equation. In g(x) = -5(x-3)^2, we can identify h = 3 and k = 0. Therefore, the vertex is located at the point (3, 0). This signifies that the parabola's highest point (maximum value) is at the coordinate (3, 0). This aligns with our understanding that the parabola opens downwards due to the negative coefficient 'a'.

Step-by-Step Guide: Unveiling the X-Intercept

To accurately determine the x-intercept of the quadratic function g(x) = -5(x-3)^2, we embark on a systematic approach. The x-intercept, as a fundamental characteristic of the parabola, reveals where the function's graph crosses the x-axis. This point is crucial for understanding the function's behavior and its solutions.

  1. Set g(x) to Zero: The journey begins by recognizing that the x-intercept occurs where the function's output, g(x), is zero. This is because the x-axis represents the line where y = 0. Therefore, we replace g(x) with 0 in the equation, transforming it into 0 = -5(x-3)^2.
  2. Isolate the Squared Term: To isolate the squared term, which holds the key to finding x, we divide both sides of the equation by -5. This step simplifies the equation, leading us closer to the solution. Dividing both sides by -5, we get 0 = (x-3)^2.
  3. Take the Square Root: With the squared term isolated, we take the square root of both sides of the equation. This action undoes the squaring, allowing us to solve for the expression inside the parentheses. Taking the square root results in 0 = x-3.
  4. Solve for x: The final step involves solving for x. By adding 3 to both sides of the equation, we isolate x and find its value. This value represents the x-coordinate of the x-intercept. Adding 3 to both sides gives us x = 3.
  5. Express as an Ordered Pair: Now that we have the x-coordinate, we express the x-intercept as an ordered pair. Remember, the x-intercept is a point on the coordinate plane, and points are represented as (x, y). Since g(x) was set to 0, the y-coordinate is 0. Thus, the x-intercept is (3, 0).

Vertex Decoded: A Step-by-Step Exploration

The vertex, another cornerstone of quadratic functions, represents the parabola's extreme point. For g(x) = -5(x-3)^2, the vertex is either the highest or lowest point on the curve, depending on whether the parabola opens upwards or downwards. Here's how we find it:

  1. Understand Vertex Form: The equation g(x) = -5(x-3)^2 is in vertex form, which is g(x) = a(x-h)^2 + k. In this form, (h, k) directly represents the vertex of the parabola. This is a powerful advantage, as it allows us to identify the vertex without further calculations.
  2. Identify h and k: By comparing g(x) = -5(x-3)^2 to the vertex form, we can easily identify the values of h and k. Here, h is 3, and k is 0. It's crucial to note that the 'h' value is the opposite of the number inside the parentheses. This is because the vertex form includes (x-h), so a positive number inside the parentheses corresponds to a positive 'h'.
  3. Express the Vertex: With h and k identified, we express the vertex as an ordered pair (h, k). In this case, the vertex is (3, 0). This means the parabola's extreme point is located at the coordinate (3, 0).

Putting It All Together: The X-Intercept and Vertex of g(x) = -5(x-3)^2

In conclusion, the x-intercept of the quadratic function g(x) = -5(x-3)^2 is the point (3, 0). This is where the parabola crosses the x-axis, signifying the real root of the equation. We found this by setting g(x) to zero and solving for x, a process that unveiled the parabola's intersection with the x-axis.

The vertex of the same function is also located at the point (3, 0). This point represents the maximum value of the function, as the parabola opens downwards due to the negative coefficient in front of the squared term. We determined this by recognizing the vertex form of the equation and extracting the h and k values, which directly correspond to the vertex coordinates.

The fact that the x-intercept and vertex are the same point in this case is a unique characteristic of this particular quadratic function. It indicates that the parabola touches the x-axis at its vertex, signifying a single real root (or a repeated root) for the equation. This is a special case that highlights the diverse behavior of quadratic functions.

By meticulously dissecting the equation g(x) = -5(x-3)^2, we have not only identified its x-intercept and vertex but also gained a deeper appreciation for the information encoded within quadratic equations. These skills are invaluable for anyone venturing further into the world of mathematics and its applications.

Practice Makes Perfect: Mastering Quadratic Function Analysis

To truly solidify your understanding of quadratic functions, it's essential to practice identifying the x-intercept and vertex in various equations. Experiment with different quadratic functions, both in vertex form and standard form, to hone your skills. Challenge yourself to sketch the parabolas based on these key features, and observe how changes in the equation affect the graph's shape and position.

Remember, mathematics is a journey of discovery, and each problem solved is a step forward. By embracing the challenge of quadratic functions, you'll unlock a powerful tool for problem-solving and a deeper appreciation for the beauty of mathematics.

The x-intercept of function g is (3, 0).

The vertex of function g is (3, 0).