Problem 22 - Calculus Volume 1

Question

For the following exercises, set up a table to sketch the graph of each function using the following values: \(x=-3,-2,-1,0,1,2,3\) $$f(x)=x^{2}+1$$

Step-by-Step Solution

Answer

Create a table with calculated values of \(f(x)\) and plot as a parabola.

Step 1: Understand the Function

The function given is a quadratic function, specifically a parabola, defined as \[f(x) = x^2 + 1\]This means that for each value of \(x\), to find \(f(x)\), we first square \(x\), then add 1 to the result.

Step 2: Create a Table of Values

We need to calculate \(f(x)\) for each given \(x\) value. We will create a table with two columns, one for \(x\) and one for \(f(x)\). The \(x\) values provided are \(-3, -2, -1, 0, 1, 2, \) and \(3\).

Step 3: Calculate f(x)

For each \(x\) value, compute \(f(x)\):- \(f(-3) = (-3)^2 + 1 = 9 + 1 = 10\)- \(f(-2) = (-2)^2 + 1 = 4 + 1 = 5\)- \(f(-1) = (-1)^2 + 1 = 1 + 1 = 2\)- \(f(0) = 0^2 + 1 = 0 + 1 = 1\)- \(f(1) = 1^2 + 1 = 1 + 1 = 2\)- \(f(2) = 2^2 + 1 = 4 + 1 = 5\)- \(f(3) = 3^2 + 1 = 9 + 1 = 10\)

Step 4: Record Calculated Values

Using the results from Step 3, fill in the table:\[\begin{array}{c|c}x & f(x) \\hline-3 & 10 \-2 & 5 \-1 & 2 \0 & 1 \1 & 2 \2 & 5 \3 & 10 \\end{array}\]

Step 5: Sketch the Graph

Plot each point from the table \((-3, 10), (-2, 5), (-1, 2), (0, 1), (1, 2), (2, 5), (3, 10)\) on a coordinate plane. Connect the points smoothly to form a parabola that opens upward. Notice the symmetry of the graph along the y-axis.

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function where the highest degree is two. It's generally expressed in the form \[f(x) = ax^2 + bx + c\], where \(a\), \(b\), and \(c\) are constants, and the term involving \(x^2\) is what makes it quadratic. For the function \(f(x) = x^2 + 1\), the equation simplifies further because there is no \(x\) or constant multiplier with \(x^2\) other than the leading one. Here:

• \(a = 1\)
• \(b = 0\)
• \(c = 1\)

The quadratic function naturally forms a U-shaped curve called a parabola when graphed. Knowing this can simplify your approach to graphing or interpreting quadratic functions. The vertex, which in this case is at the point \((0, 1)\), represents the lowest point on the parabola when the leading coefficient \(a\) is positive.

Table of Values

Creating a table of values is an invaluable method for understanding how a function behaves over a range. For the quadratic function \(f(x) = x^2 + 1\), you calculate \(f(x)\) for distinct values of \(x\) provided, such as -3 to 3. Each calculation gives us a pair of \(x\) and \(f(x)\), such as (-3, 10) or (1, 2). This forms a set of points you can plot on a graph. The values in the table reveal the function's output for each \(x\), helping us visualize the shape of the curve once plotted.

• The function enhances understanding of symmetry. Note how the table shows that \(f(-x)\) is equal to \(f(x)\), reflecting the parabola's symmetry over the y-axis, such as \(f(-1) = 2\) and \(f(1) = 2\).

Parabola

A parabola is the graph you get from a quadratic function. Its U-shape is distinctive and is characterized by its vertex and axis of symmetry. For \(f(x) = x^2 + 1\), the parabola opens upwards because the leading coefficient (1) is positive. The vertex of a parabola is its turning point, which in this context is the lowest point due to the upward opening. In this case, it’s located at \((0, 1)\). The axis of symmetry divides the parabola into two mirror images. The graph of \(f(x) = x^2 + 1\) is symmetric along the line \(x = 0\), meaning the left and right halves are mirror images. Visualizing this helps in graphing more accurately as you can confirm the positions of points and overall shape.

Coordinate Plane

In graphing functions, the coordinate plane is where all the magic happens. It consists of two axes, the horizontal \(x\)-axis and the vertical \(y\)-axis, meeting at the origin \((0,0)\). When sketching the graph of \(f(x) = x^2 + 1\), you plot the calculated points from your table of values onto this plane. Each point, such as \((-3, 10)\), corresponds to coordinates in the form \((x, f(x))\). Plot these points and draw a smooth curve through them to complete your parabola.

• This visual representation on the coordinate plane showcases the function's behavior across positive and negative values of \(x\).
• It allows you to observe attributes like symmetry and vertex position directly.

Graphing on the coordinate plane reinforces understanding of how numerical changes affect the function visually.