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Chapter 8: Problem 6

Find the coordinates of the vertex for each quadratic function listed. Then specify whether each vertex is a maximum or minimum. a. \(y=4 x^{2}\) c. \(P(n)=\left(\frac{1}{12}\right) n^{2}\) b. \(f(x)=-8 x^{2}\) d. \(Q(t)=-\left(\frac{1}{24}\right) t^{2}\)

Short Answer

Expert verified
The vertex for each function is (0, 0). Function y=4x^2 and P(n)=(1/12)n^2 have minimum vertices, while f(x)=-8x^2 and Q(t)=-(1/24)t^2 have maximum vertices.

Step by step solution

01

- Identify the vertex form of a quadratic function

The vertex form of a quadratic function is given by y = a(x-h)^2 + k where (h, k) is the vertex. For the given standard form y = ax^2, the vertex is at (0, 0).
02

- Determine the vertex for each function

For each function provided:a. For the function y = 4x^2, the vertex form is y = 4(x-0)^2 + 0, so the vertex is at (0, 0).b. For f(x) = -8x^2, the vertex form is f(x) = -8(x-0)^2 + 0, so the vertex is at (0, 0).c. For P(n) = (1/12)n^2, the vertex form is P(n) = (1/12)(n-0)^2 + 0, so the vertex is at (0, 0).d. For Q(t) = -(1/24)t^2, the vertex form is Q(t) = -(1/24)(t-0)^2 + 0, so the vertex is at (0, 0).
03

- Determine whether each vertex is a maximum or minimum

If the coefficient a is positive, the parabola opens upwards and the vertex is a minimum. If a is negative, the parabola opens downwards and the vertex is a maximum.a. For y = 4x^2, a = 4 (positive), so the vertex is a minimum.b. For f(x) = -8x^2, a = -8 (negative), so the vertex is a maximum.c. For P(n) = (1/12)n^2, a = (1/12) (positive), so the vertex is a minimum.d. For Q(t) = -(1/24)t^2, a = -(1/24) (negative), so the vertex is a maximum.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

vertex form
The vertex form of a quadratic function is incredibly useful for identifying the vertex of a parabola quickly. It is structured as: y = a(x-h)^2 + k Here,
  • a determines whether the parabola opens upwards or downwards.
  • (h, k) is the vertex of the parabola.
If you start with the standard form of a quadratic function, which is y = ax^2 + bx + c, you can convert it to vertex form by completing the square. For a simpler quadratic like y = ax^2, the vertex is always at (0, 0) because there are no linear (bx) or constant (c) terms to shift the parabola horizontally or vertically. This makes it straightforward to see that in examples like y = 4x^2 or y = -8x^2, the vertex form reveals the vertex immediately.
maximum and minimum points
Quadratic functions either have a maximum or a minimum point, which is found at the vertex. Whether a point is a maximum or a minimum depends on the value of the coefficient a in the quadratic function:
  • If a > 0, the parabola opens upwards and the vertex is a minimum point.
  • If a < 0, the parabola opens downwards and the vertex is a maximum point.
For example, in y = 4x^2, since a = 4 (positive), the vertex at (0, 0) is a minimum. On the other hand, for f(x) = -8x^2, since a = -8 (negative), the vertex at (0, 0) is a maximum point. Thus, identifying the coefficient a is crucial for determining if the vertex represents the lowest or highest point of the parabola.
quadratic functions
Quadratic functions are polynomial functions of degree 2, typically written as y = ax^2 + bx + c. They form a specific class of curves known as parabolas. Here are some of their key properties:
  • The shape of the graph is parabolic, generally resembling a U or an upside-down U.
  • The axis of symmetry passes through the vertex, splitting the parabola into two symmetrical halves.
  • The vertex represents the highest or lowest point of the parabola, determining whether it has a maximum or minimum value.
  • The direction in which the parabola opens (upwards or downwards) is controlled by the sign and value of coefficient a.
Understanding these properties helps in analyzing the behavior of quadratic functions. For example, by examining the functions y = 4x^2 and f(x) = -8x^2, we observe how the positive and negative values of a affect the direction of the parabola and the position of the vertex.

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Most popular questions from this chapter

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