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Chapter 1: Problem 17

Graph \(y=f(x)\) by hand by first plotting points to determine the shape of the graph. $$ f(x)=\frac{1}{2} x^{2} $$

Short Answer

Expert verified
The function \(f(x) = \frac{1}{2} x^2\) graphs as an upward-opening parabola with its vertex at the origin.

Step by step solution

01

Choose Values for x

Select several values for \(x\) to calculate corresponding \(y\) values. For simplicity, choose integers: \(x = -2, -1, 0, 1, 2\).
02

Calculate Corresponding y-values

Use the function \(f(x) = \frac{1}{2} x^2\) to calculate \(y\) for each chosen \(x\).\[f(-2) = \frac{1}{2}(-2)^2 = 2\] \[f(-1) = \frac{1}{2}(-1)^2 = \frac{1}{2}\] \[f(0) = \frac{1}{2}(0)^2 = 0\] \[f(1) = \frac{1}{2}(1)^2 = \frac{1}{2}\] \[f(2) = \frac{1}{2}(2)^2 = 2\] This gives the points \((-2, 2), (-1, \frac{1}{2}), (0, 0), (1, \frac{1}{2}), (2, 2)\).
03

Plot the Points on a Graph

Draw a set of axes and plot the points calculated in the previous step: \((-2, 2), (-1, \frac{1}{2}), (0, 0), (1, \frac{1}{2}), (2, 2)\).
04

Draw the Parabola

Carefully sketch a smooth curve through each of the plotted points. This will be a parabola opening upwards, with its vertex at the origin \((0, 0)\).
05

Verify Symmetry

Check the symmetry of the parabola across the \(y\)-axis by ensuring that points \((x, y)\) and \((-x, y)\) are plotted, confirming the parabolic shape.

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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 symmetrical, U-shaped curve that can open either upwards or downwards. When you graph a quadratic function like \(y = f(x) = \frac{1}{2}x^2\), the result is a parabola. In this particular case, since the value of \(x^2\) is always positive, and the coefficient \(\frac{1}{2}\) is positive, the parabola will open upwards.

Here are some key features of a parabola:
  • The parabola is defined by its curve, which is entirely smooth.
  • Each parabola has a line of symmetry that divides it into two mirror-image halves.
  • The points on one side of the symmetry line reflect over to the other side.
A parabola's nature can be altered by changing the coefficients of the equation. For instance, if you multiply \(x^2\) by a negative number, the parabola flips and opens downwards.
Vertex
The vertex is a significant point on a parabola. It represents either the maximum or minimum point depending on how the parabola opens. For \(y = \frac{1}{2}x^2\), the vertex is located at the origin point \((0, 0)\).

Characteristics of the vertex include:
  • In an upward-opening parabola, like the one for this exercise, the vertex is the lowest point.
  • In a downward-opening parabola, the vertex would be the highest point.
  • It is also situated directly on the axis of symmetry, marking the point where the parabola is symmetric.
Locating the vertex is essential when graphing a parabola as it helps anchor the curve accurately on the graph.
Symmetry
Symmetry is integral to understanding and drawing parabolas. For a parabola, the axis of symmetry is a vertical line that cuts through the vertex. In many cases, like \(y = \frac{1}{2}x^2\), it coincides with the \(y\)-axis.

Here's what to note about symmetry in parabolas:
  • Everything on one side of the symmetry line mirrors onto the other side.
  • To confirm symmetry, you can ensure that if a point \((x, y)\) is on the graph, then \((-x, y)\) should also be on the graph.
  • This characteristic helps to simplify the graphing of parabolas, as you only need to calculate and plot points on one side initially.
Using symmetry effectively allows you to graph parabolas faster and more accurately.
Plotting Points
Plotting points is a basic method to graph a function by hand. For \(y = \frac{1}{2}x^2\), we choose specific \(x\) values to start constructing the graph. Calculating corresponding \(y\) values will give us pairs of coordinates like \((-2, 2)\), \((-1, \frac{1}{2})\), and so on.

Steps in plotting points include:
  • Choose a set of simple coordinates for \(x\), such as integers, for ease in calculations.
  • Calculate \(y\) for each \(x\) using the equation \(y = \frac{1}{2}x^2\).
  • Mark each point on your graph paper, using a consistent scale for accuracy.
Once you've marked your points, draw a smooth curve through them鈥攖his forms the recognizable parabola shape. Plotting by hand is effective for visualizing functions and understanding their behavior on a graph.

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