Mr Walker's Class Claims About Quadratic Function Analysis

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Mr. Walker presented his class with the quadratic function $f(x) = (x+3)(x+5)$, sparking a lively discussion about its properties. Four students stepped forward, each making a unique claim about the function's behavior. To truly understand this function, we must carefully analyze each student's claim, dissecting the mathematical reasoning behind them and verifying their accuracy.

Understanding Quadratic Functions

Before diving into the students' claims, let's first revisit the fundamental characteristics of quadratic functions. A quadratic function is a polynomial function of degree two, generally expressed in the form $f(x) = ax^2 + bx + c$, where a, b, and c are constants and a ≠ 0. The graph of a quadratic function is a parabola, a symmetrical U-shaped curve. Key features of a parabola include its vertex (the point where the parabola changes direction), its axis of symmetry (the vertical line passing through the vertex), its x-intercepts (the points where the parabola intersects the x-axis), and its y-intercept (the point where the parabola intersects the y-axis).

In Mr. Walker's function, $f(x) = (x+3)(x+5)$, we can see that it's presented in factored form. Expanding this expression, we get $f(x) = x^2 + 8x + 15$. This form reveals the coefficients a = 1, b = 8, and c = 15, which will be useful in determining various characteristics of the function.

Examining Student Claims

Now, let's delve into the claims made by the four students and evaluate their validity. Each claim touches upon a specific aspect of the quadratic function, requiring us to apply our knowledge of quadratic properties and algebraic techniques.

I. Jeremiah's Claim The Y-Intercept

Jeremiah asserts that the y-intercept of the function is at (15, 0). To assess this claim, we need to recall what a y-intercept represents. The y-intercept is the point where the graph of the function intersects the y-axis. This occurs when x = 0. To find the y-intercept, we substitute x = 0 into the function and solve for f(x).

Substituting x = 0 into $f(x) = (x+3)(x+5)$, we get:

f(0)=(0+3)(0+5)=3∗5=15f(0) = (0+3)(0+5) = 3 * 5 = 15

Therefore, the y-intercept is at the point (0, 15), not (15, 0) as Jeremiah claimed. Jeremiah's claim is incorrect. He seems to have confused the y-coordinate with the x-coordinate. Understanding intercepts is crucial in grasping the behavior of functions, including quadratics.

II. Lindsay's Claim The X-Intercepts

Lindsay claims that the x-intercepts are at (-3, 0) and (-5, 0). The x-intercepts, also known as the roots or zeros of the function, are the points where the graph intersects the x-axis. At these points, f(x) = 0. To find the x-intercepts, we set the function equal to zero and solve for x.

Using the factored form of the function, $f(x) = (x+3)(x+5)$, we set it equal to zero:

(x+3)(x+5)=0(x+3)(x+5) = 0

This equation holds true if either factor is equal to zero. Thus, we have two possible solutions:

  • x + 3 = 0 => x = -3
  • x + 5 = 0 => x = -5

Therefore, the x-intercepts are indeed at (-3, 0) and (-5, 0), confirming Lindsay's claim. Lindsay's claim is correct. The factored form of the quadratic function makes it easy to identify the x-intercepts, as they are the values of x that make each factor equal to zero. This highlights the importance of understanding the different forms of quadratic equations and their respective advantages.

III. Michael's Claim The Vertex

Michael claims that the vertex of the parabola is at (-4, -1). The vertex is a crucial point on the parabola, representing either its minimum value (if the parabola opens upwards) or its maximum value (if the parabola opens downwards). To find the vertex, we can use a couple of methods. One method involves finding the axis of symmetry, which is the vertical line that passes through the vertex. The x-coordinate of the vertex lies on the axis of symmetry, and it can be found using the formula: x = -b / 2a, where a and b are the coefficients in the standard form of the quadratic equation ($f(x) = ax^2 + bx + c$).

In our case, a = 1 and b = 8. So, the x-coordinate of the vertex is:

x = -8 / (2 * 1) = -4

Now, to find the y-coordinate of the vertex, we substitute x = -4 into the function:

f(−4)=(−4+3)(−4+5)=(−1)(1)=−1f(-4) = (-4+3)(-4+5) = (-1)(1) = -1

Therefore, the vertex is indeed at (-4, -1), confirming Michael's claim. Michael's claim is correct. Finding the vertex is essential for understanding the parabola's extreme point and its overall shape. The formula x = -b / 2a is a powerful tool for determining the axis of symmetry and subsequently the x-coordinate of the vertex.

IV. Natalie's Claim The Axis of Symmetry

Natalie claims that the axis of symmetry is the line x = -4. The axis of symmetry is a vertical line that divides the parabola into two symmetrical halves. As mentioned in the previous section, the axis of symmetry passes through the vertex. We already calculated the x-coordinate of the vertex to be -4. Therefore, the equation of the axis of symmetry is indeed x = -4, validating Natalie's claim.

Natalie's claim is correct. The axis of symmetry is a fundamental property of a parabola, and understanding its relationship to the vertex is crucial. Knowing the axis of symmetry allows us to easily identify symmetrical points on the parabola and visualize its overall shape.

Conclusion

In conclusion, after carefully analyzing the claims made by Mr. Walker's students, we found that Lindsay, Michael, and Natalie made correct statements about the function $f(x) = (x+3)(x+5)$. Lindsay correctly identified the x-intercepts, Michael accurately determined the vertex, and Natalie correctly stated the axis of symmetry. However, Jeremiah's claim about the y-intercept was incorrect. This exercise highlights the importance of understanding the key features of quadratic functions and applying algebraic techniques to verify their properties. By dissecting each claim, we've reinforced our understanding of intercepts, vertices, and axes of symmetry, all crucial components in the study of quadratic functions. This analysis not only clarifies the specific characteristics of the given function but also provides a framework for analyzing other quadratic functions in the future. The ability to accurately identify these features is fundamental in various mathematical applications and real-world scenarios where quadratic models are employed.

Key Takeaways

  • Y-intercept: The point where the graph intersects the y-axis (x=0). Jeremiah's mistake highlights the importance of accurate substitution and interpretation of results.
  • X-intercepts: The points where the graph intersects the x-axis (f(x)=0). Lindsay's correct identification demonstrates the usefulness of the factored form in finding roots.
  • Vertex: The minimum or maximum point of the parabola. Michael's accurate calculation showcases the utility of the formula x = -b/2a.
  • Axis of Symmetry: The vertical line that divides the parabola into two symmetrical halves. Natalie's correct statement emphasizes the relationship between the vertex and the axis of symmetry.

By mastering these concepts, students can confidently analyze and interpret quadratic functions, paving the way for more advanced mathematical explorations. Understanding quadratic functions is not just an academic exercise; it's a foundational skill with applications in physics, engineering, economics, and many other fields. From projectile motion to optimization problems, quadratic models provide valuable insights into real-world phenomena. Therefore, a thorough grasp of their properties is essential for students pursuing various STEM disciplines and beyond. The dialogue in Mr. Walker's class serves as a model for how students can actively engage with mathematical concepts, challenge each other's thinking, and collaboratively build a deeper understanding. Such interactive learning environments foster critical thinking skills and promote a more meaningful appreciation for the beauty and power of mathematics.

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  • Quadratic Functions
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  • Vertex
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  • Factored Form
  • Standard Form
  • Algebra
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This article analyzes claims made about the quadratic function f(x) = (x+3)(x+5), discussing x-intercepts, y-intercepts, the vertex, and the axis of symmetry.