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Chapter 2: Problem 13

Sketch the graph of the function by first making a table of values. $$h(x)=16-x^{2}$$

Short Answer

Expert verified
Plot points from table and connect to form a downward-opening parabola.

Step by step solution

01

Understand the Function

The given function is \( h(x) = 16 - x^2 \). This is a quadratic function which represents a parabola. The parabola opens downwards because the coefficient of \( x^2 \) is negative (-1). The vertex of this parabola is the highest point.
02

Find the Vertex

The standard form of the quadratic function is \( h(x) = ax^2 + bx + c \). In our function, \( a = -1 \), \( b = 0 \), and \( c = 16 \). The vertex \( x \)-coordinate is given by \( x = -\frac{b}{2a} \). Here, it simplifies to \( x = 0 \). Hence, the vertex is at \( (0, 16) \).
03

Create a Table of Values

Choose a set of \( x \)-values around the vertex, typically from \( -3 \) to \( 3 \) for simplicity. Calculate \( h(x) \) for these values:- When \( x = -3, h(x) = 16 - (-3)^2 = 7 \)- When \( x = -2, h(x) = 16 - (-2)^2 = 12 \)- When \( x = -1, h(x) = 16 - (-1)^2 = 15 \)- When \( x = 0, h(x) = 16 - 0^2 = 16 \)- When \( x = 1, h(x) = 16 - 1^2 = 15 \)- When \( x = 2, h(x) = 16 - 2^2 = 12 \)- When \( x = 3, h(x) = 16 - 3^2 = 7 \).
04

Sketch the Graph

Using the table of values, plot the points \((-3, 7)\), \((-2, 12)\), \((-1, 15)\), \((0, 16)\), \((1, 15)\), \((2, 12)\), and \((3, 7)\) on a coordinate plane. Connect these points with a smooth, downward-opening parabola. The symmetry about the \( y \)-axis should be evident.

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Key Concepts

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

Quadratic Function
A quadratic function is a type of polynomial that can be expressed in the form \( f(x) = ax^2 + bx + c \). It's called a "quadratic" because it deals with squares, as the term \( x^2 \) is the highest power of the variable \( x \). This results in a U-shaped graph known as a parabola. In any quadratic function:
  • \( a \) is the coefficient of \( x^2 \). It determines the direction the parabola opens. A positive \( a \) means it opens upwards, and a negative \( a \) makes it open downwards.
  • \( b \) and \( c \) affect the position of the parabola on the graph.
In our function \( h(x) = 16 - x^2 \):
  • \( a = -1 \), causing the parabola to open downwards.
  • \( b = 0 \), indicating the parabola is symmetric about the \( y \)-axis, or the line \( x = 0 \).
  • \( c = 16 \), placing the highest point of the parabola at a height of 16 on the \( y \)-axis.
Understanding these parameters helps in predicting the general shape and position of the parabola before graphing.
Vertex of Parabola
The vertex of a parabola is a crucial point that indicates the maximum or minimum value of the quadratic function. For a parabola that opens downwards, like in our example, the vertex is at the maximum point.
The vertex can be found using the formula for the \( x \)-coordinate: \( x = -\frac{b}{2a} \).
  • For \( h(x) = 16 - x^2 \), we have \( a = -1 \) and \( b = 0 \).
  • Substituting these into the formula gives \( x = -\frac{0}{-2} = 0 \).
  • To find the \( y \)-coordinate, substitute \( x = 0 \) back into the function: \( h(0) = 16 - 0^2 = 16 \).
Therefore, the vertex of our parabola is at \( (0, 16) \).
This point is where the parabola reaches its peak.
Knowing how to find the vertex is essential for graphing because it serves as a central point for the parabola's symmetry.
Table of Values
Creating a table of values is a helpful technique to visually understand how a quadratic function behaves over a range of \( x \)-values. By calculating \( h(x) \) for various \( x \)-values:
  • You can see how the function rises and falls, giving insight into its shape.
  • Points of symmetry relative to the vertex become evident.
For the function \( h(x) = 16 - x^2 \), we calculate values near the vertex.
  • Choose a range of \( x \) such as from \(-3\) to \(3\).
  • Calculate \( h(x) \) for each value: \((-3, 7), (-2, 12), (-1, 15), (0, 16), (1, 15), (2, 12), (3, 7)\).
These points reveal the symmetrical nature of the parabola about the \( y \)-axis.
Once made, you can plot these points on a graph,
which creates the smooth, curved shape typical of a parabola.
This approach simplifies the graphing process and ensures accuracy.

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