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Chapter 24: Problem 43

Sketch the indicated curves by the methods of this section. You may check the graphs by using a calculator. A batter hits a baseball that follows a path given by \(y=x-0.0025 x^{2},\) where distances are in feet. Sketch the graph of the path of the baseball.

Short Answer

Expert verified
The baseball follows a downward-opening parabola with vertex at (200, 100).

Step by step solution

01

Identify the Curve Type

The equation is a quadratic equation in the form of \(y = ax^2 + bx + c\). In this equation, \(a = -0.0025\), \(b = 1\), and \(c = 0\). Since \(a < 0\), it describes a downward-opening parabola. This identifies the curve as a parabola that opens downward.
02

Determine Key Features

To understand the shape of this parabola, identify key features: the vertex, axis of symmetry, and the direction it opens. The axis of symmetry for the equation \(y = ax^2 + bx + c\) is given by \(x = -\frac{b}{2a}\). Substituting the given \(a\) and \(b\), we get \(x = -\frac{1}{2(-0.0025)} = 200\). The vertex is the maximum point because the parabola opens downward.
03

Calculate the Vertex

Now substitute \(x = 200\) into the original equation to find the \(y\)-coordinate of the vertex. The vertex is at \(y = 200 - 0.0025(200)^2 = 200 - 100 = 100\). So the vertex of the parabola is \((200, 100)\).
04

Identify Additional Points

Choose additional points on either side of the vertex to help sketch the graph. Select values for \(x\), such as \(0\) and \(400\), to find corresponding \(y\)-values. For \(x = 0\), \(y = 0\). For \(x = 400\), \(y = 0\), confirming it passes through the origin and symmetrically at \(x = 400\).
05

Sketch the Parabola

Now, plot the points \((0,0)\), \((200,100)\), and \((400,0)\) on a coordinate system. Draw a smooth curve through these points forming a downward-opening parabola, ensuring symmetry about \(x = 200\), which is the axis of symmetry. This curve represents the path of the baseball.
06

Double-Check with Calculator

To verify, use a graphing calculator to enter the equation \(y = x - 0.0025x^2\). Confirm that the graph matches the sketch with a vertex at \((200, 100)\) and intercepts at \((0,0)\) and \((400,0)\).

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Key Concepts

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

Parabola
A parabola is a U-shaped curve that appears in the graph of a quadratic equation. Parabolas can open upward or downward, depending on the sign of the coefficient in the quadratic term. In the quadratic equation of the form \(y = ax^2 + bx + c\), the value of \(a\) determines the direction the parabola opens.
  • If \(a > 0\), the parabola opens upward like a U.
  • If \(a < 0\), it opens downward like an upside-down U, as in our baseball path example.
Parabolas are symmetric and have important features such as a vertex and axis of symmetry which help in graphing and understanding their properties. Understanding these features will help in sketching and interpreting the graph accurately.
Vertex
The vertex is the peak or the lowest point of a parabola depending on its orientation. It represents the point where the parabola changes direction from increasing to decreasing, or vice versa. For a downward-opening parabola like in the example, the vertex is the highest point.
To find the vertex in a quadratic equation \(y = ax^2 + bx + c\):
  • Use the formula to find the x-coordinate: \(x = -\frac{b}{2a}\).
  • Substitute this \(x\)-value back into the original equation to find the \(y\)-coordinate.
In our example, the vertex is at \((200, 100)\), signifying this is the highest point of the baseball's path.
Axis of Symmetry
An axis of symmetry is a vertical line that divides a parabola into two mirror-image halves. Every point on one side of the axis has a corresponding point on the other side, helping in graphing the parabola easily.
For a quadratic equation \(y = ax^2 + bx + c\), the axis of symmetry can be found using the formula:
  • \(x = -\frac{b}{2a}\) which is the same as the x-coordinate of the vertex.
In our baseball path equation, the axis of symmetry is \(x = 200\). This line indicates that for each point on one side of the parabola, there is an identical point on the opposite side.
Downward-Opening Parabola
A downward-opening parabola is characterized by its maximum point at the vertex and opening downward from there. This happens when the leading coefficient \(a\) in the quadratic equation is negative.
In this orientation:
  • The vertex is the highest point on the graph.
  • The graph is symmetrical about the axis of symmetry.
  • Such parabolas often model situations where an object has a peak point, like the path of a thrown ball.
In our exercise, the parabola describes the baseball's path, peaking at \((200, 100)\) before descending symmetrically.

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Sketch the indicated curves by the methods of this section. You may check the graphs by using a calculator. The angle \(\theta\) (in degrees) of a robot arm with the horizontal as a function of the time \(t\) (in \(s\) ) is given by \(\theta=10+12 t^{2}-2 t^{3}\) Sketch the graph for \(0 \leq t \leq 6\).

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