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Chapter 5: Problem 16

Write down the given quadratic function on your homework paper, then state the coordinates of the vertex. $$f(x)=-6(x+8)^{2}+1$$

Short Answer

Expert verified
The vertex coordinates are (-8, 1).

Step by step solution

01

Identify the Quadratic Formula

The given quadratic function is in vertex form: \( f(x) = a(x - h)^2 + k \). In this particular example, \( f(x) = -6(x+8)^2 + 1 \).
02

Note the Vertex Formula Components

The vertex form \( f(x) = a(x-h)^2+k \) includes the elements \( h \) and \( k \) which represent the coordinates of the vertex \((h, k)\). Here, \( h = -8 \) because it's \( x - (-8) \), and \( k = 1 \).
03

State the Vertex Coordinates

From the function \( f(x) = -6(x+8)^2 + 1 \), we identify \( h = -8 \) and \( k = 1 \). Thus, the coordinates of the vertex are \((-8, 1)\).

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Key Concepts

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

vertex form of a quadratic function
The vertex form of a quadratic function is one of the most intuitive ways to express a quadratic equation. It is written as \( f(x) = a(x-h)^2 + k \). This format highlights the vertex of the parabola, making it easy to find its coordinates. In this form, \( a \), \( h \), and \( k \) have specific roles:
  • \( a \) determines the direction of the parabola (upwards for positive \( a \), downwards for negative \( a \)) and its width, with larger absolute values making the parabola narrower.
  • \( h \) is the x-coordinate of the vertex, and is derived from \((x-h)^2\), which indicates a horizontal shift.
  • \( k \) is the y-coordinate of the vertex, indicating a vertical shift.
By arranging the quadratic in this form, we can effortlessly spot modifications to the graph of the function. Converting a quadratic function into vertex form enables easy graphing and a straightforward interpretation of the graph's key features.
quadratic function
A quadratic function is a fundamental concept in algebra and can be described as any function that can be rewritten in the form \( f(x) = ax^2 + bx + c \). Quadratic functions create a parabolic graph, a U-shaped curve that opens either upward or downward depending on the sign of \( a \).
Key characteristics of a quadratic function include:
  • The vertex, which is the highest or lowest point of the parabola.
  • The axis of symmetry, a vertical line that passes through the vertex, can be useful in graphing as it reflects the function's shape on either side.
  • The direction of the parabola, indicated by the sign of \( a \); if \( a > 0 \), the parabola opens upwards, while if \( a < 0 \), it opens downwards.
Understanding these elements is crucial, as they directly affect the graph and properties of the function. Quadratic functions frequently appear in a variety of contexts, such as physics and economics, due to their ability to model complex relationships.
coordinates of the vertex
The coordinates of the vertex are a pivotal feature of any quadratic function expressed in vertex form. As seen in this exercise, the vertex provides a clear point of reference for describing the behavior of the parabola formed by a quadratic function.
For a function given in the form \( f(x) = a(x-h)^2 + k \), the vertex is located at \((h, k)\). In simple terms, the vertex tells us:
  • Position: The point \((h, k)\) tells us exactly where the highest or lowest point of the parabola is located.
  • Symmetry: The x-coordinate \( h \) of the vertex coincides with the axis of symmetry. This means that the parabola mirrors itself across this vertical line.
  • Direction: Relative to \( k \), it helps to determine whether the vertex is a maximum or minimum point (affected by the sign of \( a \)).
Finding these coordinates is often the first step in graphing or analyzing a quadratic function's behavior. Precise determination of the vertex enhances understanding of the graph structure and assists in problem-solving contexts, such as finding maximum or minimum values in practical applications.

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Most popular questions from this chapter

Write down the given quadratic function on your homework paper, then state the coordinates of the vertex. $$f(x)=-\frac{2}{9}\left(x+\frac{2}{3}\right)^{2}-\frac{4}{5}$$

Use your graphing calculator to sketch the graphs of \(f(x)=x^{2}, g(x)=2 x^{2}\), and \(h(x)=4 x^{2}\) on one screen. Write a short sentence explaining what you learned in this exercise.

Perform each of the following tasks for the given quadratic function. 1\. Set up a coordinate system on graph paper. Label and scale each axis. 2\. Plot the vertex of the parabola and label it with its coordinates. 3\. Draw the axis of symmetry and label it with its equation. 4\. Set up a table near your coordinate system that contains exact coordinates of two points on either side of the axis of symmetry. Plot them on your coordinate system and their "mirror images" across the axis of symmetry. 5\. Sketch the parabola and label it with its equation. 6\. Use interval notation to describe both the domain and range of the quadratic function. $$f(x)=(x-3)^{2}$$

Write the given quadratic function on your homework paper, then use set- builder and interval notation to describe the domain and the range of the function. $$f(x)=-7(x+5)^{2}-7$$

Perform each of the following tasks for the given quadratic function. 1\. Set up a coordinate system on graph paper. Label and scale each axis. 2\. Plot the vertex of the parabola and label it with its coordinates. 3\. Draw the axis of symmetry and label it with its equation. 4\. Set up a table near your coordinate system that contains exact coordinates of two points on either side of the axis of symmetry. Plot them on your coordinate system and their "mirror images" across the axis of symmetry. 5\. Sketch the parabola and label it with its equation. 6\. Use interval notation to describe both the domain and range of the quadratic function. $$f(x)=-\frac{1}{2}(x+1)^{2}+5$$
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