Analyzing The Quadratic Function G(x) = 10x^2 - 100x + 213 In Vertex Form

In the realm of mathematics, quadratic functions hold a position of paramount importance, frequently manifesting in diverse applications across fields such as physics, engineering, and economics. Among the various forms of representing quadratic functions, the vertex form stands out for its ability to readily reveal key characteristics of the function's graph, including its vertex, axis of symmetry, and direction of opening. This comprehensive guide delves into the intricacies of the vertex form and its application in analyzing the quadratic function $g(x) = 10x^2 - 100x + 213$. By transforming the function into its vertex form, $g(x) = 10(x - 5)^2 - 37$, we can unlock a wealth of information about its behavior and graphical representation.

The vertex form of a quadratic function is expressed as $g(x) = a(x - h)^2 + k$, where:

• 'a' dictates the direction and steepness of the parabola.
• '(h, k)' represents the coordinates of the vertex, the point where the parabola changes direction.

In our case, the function $g(x) = 10(x - 5)^2 - 37$ is already presented in vertex form. By carefully examining this form, we can readily extract the following information:

• The value of 'a' is 10, indicating that the parabola opens upwards (since 'a' is positive) and is relatively steep.
• The vertex is located at the point (5, -37), signifying the minimum value of the function.

The axis of symmetry is a vertical line that gracefully divides the parabola into two symmetrical halves. It elegantly passes through the vertex, serving as a mirror that reflects the parabola's shape. The equation of the axis of symmetry is given by $x = h$, where 'h' is the x-coordinate of the vertex.

In our example, the vertex is (5, -37), so the axis of symmetry is the vertical line $x = 5$. This line acts as a perfect mirror, reflecting the parabola's left side onto its right side, and vice versa. Understanding the axis of symmetry is crucial for visualizing the parabola's symmetry and predicting its behavior.

The coefficient 'a' in the vertex form plays a pivotal role in shaping the parabola's appearance. It governs two key characteristics: the direction of opening and the steepness of the curve.

• Direction of Opening:
  • If 'a' is positive, the parabola opens upwards, resembling a gentle smile. This indicates that the vertex represents the minimum point of the function.
  • If 'a' is negative, the parabola opens downwards, resembling a frown. In this case, the vertex represents the maximum point of the function.
• Steepness:
  • The absolute value of 'a' determines the parabola's steepness. A larger absolute value of 'a' results in a steeper parabola, while a smaller absolute value leads to a gentler, wider curve.

In our function, $g(x) = 10(x - 5)^2 - 37$, the value of 'a' is 10, which is positive. This confirms that the parabola opens upwards, and the vertex (5, -37) represents the minimum point of the function. Furthermore, the relatively large value of 'a' indicates that the parabola is quite steep.

For a parabola that opens upwards, the vertex represents the minimum point of the function. The y-coordinate of the vertex corresponds to the minimum value of the function. In our case, the vertex is (5, -37), so the minimum value of the function is -37. This means that no matter what value we substitute for 'x' in the function, the result will never be less than -37. The minimum value provides a crucial benchmark for understanding the function's range and overall behavior.

Now that we have thoroughly analyzed the function $g(x) = 10(x - 5)^2 - 37$, let's evaluate the given statements:

A. The axis of symmetry is the line $x = -5$.

This statement is incorrect. As we established earlier, the axis of symmetry is the vertical line that passes through the vertex. In this case, the vertex is (5, -37), so the axis of symmetry is $x = 5$, not $x = -5$.

B. The value of 'a', when the equation is written in vertex form, is 10.

This statement is correct. The vertex form of the function is $g(x) = 10(x - 5)^2 - 37$, and the coefficient 'a' is indeed 10. This value dictates the parabola's direction (upwards) and steepness.

C. The graph of $g(x)$ has a minimum at $(-37)$.

This statement is partially correct but requires clarification. The graph of $g(x)$ does have a minimum, but the minimum value is -37, which occurs at the vertex (5, -37). The statement should ideally specify that the minimum value is -37.

D. The graph of $g(x)$ has a minimum at $(5, -37)$.

This statement is correct. The vertex of the parabola, (5, -37), represents the minimum point on the graph. Since the parabola opens upwards, this is the lowest point the graph reaches.

E. The axis of symmetry is the line $x=5$.

This statement is correct. As we determined earlier, the axis of symmetry is the vertical line that passes through the vertex, which is $x = 5$.

In this comprehensive guide, we have explored the power of the vertex form in analyzing quadratic functions. By transforming the function $g(x) = 10x^2 - 100x + 213$ into its vertex form, $g(x) = 10(x - 5)^2 - 37$, we have successfully identified its vertex, axis of symmetry, direction of opening, and minimum value. This knowledge allows us to accurately describe the function's behavior and graphical representation.

The vertex form serves as a valuable tool for understanding quadratic functions and their applications. By mastering this form, we can readily extract key information and solve a wide range of problems involving quadratic equations and parabolas. Remember, the vertex form is your key to unlocking the secrets of quadratic functions!

Vertex form, quadratic function, axis of symmetry, minimum value, parabola, vertex, direction of opening, steepness, $g(x) = 10x^2 - 100x + 213$, $g(x) = 10(x - 5)^2 - 37$

Q: What is the vertex form of a quadratic function? A: The vertex form of a quadratic function is $g(x) = a(x - h)^2 + k$, where (h, k) is the vertex and 'a' determines the direction and steepness.

Q: How do you find the axis of symmetry from the vertex form? A: The axis of symmetry is the vertical line $x = h$, where 'h' is the x-coordinate of the vertex.

Q: What does the value of 'a' tell you about the parabola? A: The value of 'a' indicates the direction of opening (upwards if positive, downwards if negative) and the steepness of the parabola.

Q: How do you find the minimum or maximum value of a quadratic function in vertex form? A: The minimum or maximum value is the y-coordinate of the vertex, 'k'. If the parabola opens upwards, it's a minimum; if it opens downwards, it's a maximum.

Q: What are the correct statements about $g(x) = 10(x - 5)^2 - 37$? A: The correct statements are: B. The value of 'a' is 10, D. The graph has a minimum at (5, -37), and E. The axis of symmetry is the line x = 5.