Problem 18 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

Graph a parabola passing through the points: \((-2,8), (-1,2), (0,0), (1,2), (2,8)\).

Step 1: Choose Values for x

To start graphing the function, select various values for \( x \) to substitute into the function \( f(x) = 2x^2 \). Common choices are integer values such as \( -2, -1, 0, 1, \) and \( 2 \).

Step 2: Calculate f(x) for Each x Value

Substitute the chosen \( x \) values into the function \( f(x) = 2x^2 \). \[ f(-2) = 2(-2)^2 = 8 \]\[ f(-1) = 2(-1)^2 = 2 \]\[ f(0) = 2(0)^2 = 0 \]\[ f(1) = 2(1)^2 = 2 \]\[ f(2) = 2(2)^2 = 8 \]

Step 3: Plot the Points on a Graph

Using the calculated points, plot them on a Cartesian plane. The points are \((-2, 8), (-1, 2), (0, 0), (1, 2),\) and \((2, 8)\).

Step 4: Sketch the Graph

Connect the plotted points with a smooth curve to illustrate the shape of the graph. Since \( f(x) = 2x^2 \) is a quadratic function, the graph will be a parabola opening upwards.

Key Concepts

Plotting Points

Plotting points is the foundational step in graphing functions. It involves selecting specific values for a variable, often denoted as \(x\), and calculating corresponding outcomes or \(y\)-values using the given function. In this case, the function is \(f(x) = 2x^2\). To begin, choose several values for \(x\). It's common to pick a range that includes negative, zero, and positive numbers, such as \(-2, -1, 0, 1,\) and \(2\).
Substitute these \(x\)-values into the function to find \(f(x)\). This results in points like \((-2, 8), (-1, 2), (0, 0), (1, 2),\) and \((2, 8)\). Each point is an \(( x, y )\) coordinate that you can plot on a graph.
Remember:

• Choose a sufficient number of points to get a clear picture of the function's behavior.
• Ensuring a mix of negative and positive \(x\)-values helps in understanding how the function behaves around the origin.

Cartesian Plane

The Cartesian plane is a two-dimensional space used to visualize and analyze mathematical functions. It is divided into four quadrants by the horizontal \(x\)-axis and the vertical \(y\)-axis. These axes intersect at the origin, the point \((0, 0)\).
When plotting points on this plane, you place each point at the coordinates \((x, y)\) based on your calculations. For instance, the point \((-2, 8)\) means moving two units left of the origin on the \(x\)-axis and eight units up on the \(y\)-axis.
Here are some key points to remember:

• Positive \(x\)-values are plotted to the right of the origin, while negative \(x\)-values go to the left.
• Similarly, positive \(y\)-values go above the origin, and negative \(y\)-values are below it.
• The Cartesian plane is a helpful tool for visualizing various types of functions, including linear, quadratic, and more complex curves.

Parabola

A parabola is a specific type of graph that represents a quadratic function, such as \(f(x) = 2x^2\). This shape is characterized by its U-like appearance, opening either upwards or downwards.
In our exercise, we determined that the parabola opens upwards based on the positive coefficient of the \(x^2\) term. Connecting the plotted points with a smooth curve forms this parabola's distinct shape.
Some important aspects of parabolas:

• The vertex is the lowest or highest point of the parabola. For \(f(x) = 2x^2\), the vertex is at \((0, 0)\).
• The axis of symmetry is a vertical line passing through the vertex, here it's \(x = 0\).
• Parabolas may widen or narrow based on the coefficient of \(x^2\); a larger positive coefficient indicates a steeper curve.

Understanding these key characteristics helps in predicting how quadratic functions behave visually.