Problem 20 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

The graph of \(f(x) = x^2 + 1\) is a parabola opening upwards with a vertex at (0, 1).

Step 1: Choose Values of x

Select a range of x-values to use for plotting the graph. A good range for quadratic functions often spans from -3 to 3 to capture the critical points, turning points, and symmetry characteristic. Let's choose x-values: -3, -2, -1, 0, 1, 2, 3.

Step 2: Calculate Corresponding y-Values

For each chosen x-value, substitute it into the function to find the corresponding y-value.- For \(x = -3\), \(f(-3) = (-3)^2 + 1 = 9 + 1 = 10\).- For \(x = -2\), \(f(-2) = (-2)^2 + 1 = 4 + 1 = 5\).- For \(x = -1\), \(f(-1) = (-1)^2 + 1 = 1 + 1 = 2\).- For \(x = 0\), \(f(0) = 0^2 + 1 = 0 + 1 = 1\).- For \(x = 1\), \(f(1) = 1^2 + 1 = 1 + 1 = 2\).- For \(x = 2\), \(f(2) = 2^2 + 1 = 4 + 1 = 5\).- For \(x = 3\), \(f(3) = 3^2 + 1 = 9 + 1 = 10\).

Step 3: Plot the Points

Now plot the calculated points (x, y) on a coordinate plane: - (-3, 10) - (-2, 5) - (-1, 2) - (0, 1) - (1, 2) - (2, 5) - (3, 10) Make sure the points are accurately placed on the graph based on their coordinates.

Step 4: Draw the Curve

Connect the plotted points smoothly to form a curve. Since \(y = x^2 + 1\) is a quadratic function, the graph will be a parabola opening upwards. Ensure the curve passes through each plotted point and maintains symmetry about the y-axis.

Step 5: Verify the Graph's Symmetry and Shape

Check that the graph has the expected shape of a parabola opening upwards with a vertex at the point (0, 1). Verify that the graph is symmetric around the y-axis and that points mirror each other across the axis.

Key Concepts

Plotting Points

One fundamental step in graphing quadratic functions is plotting points. This involves selecting specific x-values and determining their corresponding y-values based on the given function.
Doing this helps us shape the graph accurately. Typically, choosing a symmetric range around zero, like from -3 to 3, can be very effective. This ensures we can see any turning points or critical features of the quadratic function.
The function given, \(f(x) = x^2 + 1\), can be evaluated by substituting the x-values to find the y-values. For instance, substituting

• \(x = -3\) gives \(f(-3) = 10\).
• For \( x = -2\), \(f(-2) = 5\).
• Similarly, for \(x = 0\), \(f(0) = 1\), which is a critical point as it's the vertex of the parabola.

Once you have a set of points like (-3, 10) and (0, 1), plot them on a graph to visualize the curve's pattern.

Symmetry in Functions

Symmetry plays a crucial role in understanding the behavior of quadratic functions. Quadratics inherently exhibit symmetry, particularly around their vertex. In the function \(f(x) = x^2 + 1\), symmetry is apparent due to the constant term and the parabolic nature.
A parabola's symmetry means if you were to fold the graph along a vertical line—a vertical line through the vertex (y-axis for this function) would mean the left side mirrors the right side. This is why when plotting points, the y-values will repeat in a mirrored fashion:

• Points like (-2, 5) and (2, 5) reflect over the y-axis.
• The graph passes through (0, 1), indicating its vertex and axis of symmetry at \(x=0\).

Understanding this symmetry helps to sketch approximate positions before plotting every point, providing a quick visual guide.

Parabolas

A parabolic graph is the visual representation of a quadratic function, with a unique "U" shape, either opening upwards or downwards depending on the sign of the quadratic term. In the equation given, \(y = x^2 + 1\), the positive coefficient of \(x^2\) indicates the parabola opens upwards.
Key characteristics of parabolas include:

• The **vertex**, which is the highest or lowest point, serving as the turning point of the function.
• The **axis of symmetry**, a vertical line passing through the vertex, dividing the parabola perfectly in half.
• **Openness**, determined by the sign of the \(x^2\) term; positive coefficient means open upwards.

When graphing, once you plot points such as (0, 1) and observe the symmetric pairs like (-3, 10) and (3, 10), a clear visual of a parabola emerges.
Drawing a smooth curve through these points while respecting their symmetry ensures you accurately capture the essence of the parabola's shape.