Problem 12 - Algebra and Trigonometry

Question

Sketch the graph of the function by first making a table of values. \(f(x)=x^{2}-4\)

Step-by-Step Solution

Answer

The graph of \(f(x) = x^2 - 4\) is a parabola opening upwards with vertex at \((0, -4)\).

Step 1: Choose Values for x

To begin, we need to select a range of x-values to make a table. For simplicity, let's choose: \(-3, -2, -1, 0, 1, 2,\) and \(3\). You can choose more values if desired for a more detailed graph.

Step 2: Calculate f(x) for Each x

Next, calculate \(f(x) = x^2 - 4\) for each selected \(x\) value.\(\)- For \(x = -3\), \(f(x) = (-3)^2 - 4 = 5\).\(\)- For \(x = -2\), \(f(x) = (-2)^2 - 4 = 0\).\(\)- For \(x = -1\), \(f(x) = (-1)^2 - 4 = -3\).\(\)- For \(x = 0\), \(f(x) = 0^2 - 4 = -4\).\(\)- For \(x = 1\), \(f(x) = 1^2 - 4 = -3\).\(\)- For \(x = 2\), \(f(x) = 2^2 - 4 = 0\).\(\)- For \(x = 3\), \(f(x) = 3^2 - 4 = 5\).

Step 3: Create a Table of Values

Now, make a table in which you list each \(x\) value alongside its corresponding \(f(x)\) value from Step 2.\[\begin{array}{c|c} x & f(x) \\hline-3 & 5 \-2 & 0 \-1 & -3 \0 & -4 \1 & -3 \2 & 0 \3 & 5 \\end{array}\]

Step 4: Plot Points on the Graph

Using the table from Step 3, plot each pair \((x, f(x))\) on a coordinate plane. This gives the points: \((-3, 5), (-2, 0), (-1, -3), (0, -4), (1, -3), (2, 0), (3, 5)\).

Step 5: Draw the Graph

After plotting the points, connect them smoothly, drawing a curved line that passes through all the points. The graph will form a parabola opening upwards, with the vertex at the lowest point, \((0, -4)\).

Key Concepts

Graphing Parabolas

Graphing a parabola is a visual way to understand quadratic functions such as \( f(x) = x^2 - 4 \). Parabolas have distinctive U-shaped curves that either open upwards or downwards. In this case, our parabola opens upwards as the coefficient of \( x^2 \) is positive.
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To graph a parabola, start by plotting several points based on calculated values of \( f(x) \) from different \( x \) values. Points plotted on a graph create a pattern that outlines the shape of the parabola.
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Here's a list to remember while graphing parabolas:

• They are symmetrical about the vertical line passing through the vertex.
• The direction of the opening depends on the sign of the \( x^2 \) coefficient.
• The vertex provides the parabola's highest or lowest point, depending on its opening.

Connecting these points smoothly will give a precise graph of the quadratic function, emphasizing the curve's symmetry.

Table of Values

Creating a table of values is a step-by-step method to help plot the graph of a function. For the function \( f(x) = x^2 - 4 \), a table of values helps us find specific pairs of \( x \) and \( f(x) \). These pairs form the basis of our plotting process.
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Here's why a table of values is helpful:

• It provides organized data, displaying corresponding \( x \) and \( f(x) \) values clearly.
• The table helps identify patterns in values, showing how \( f(x) \) changes as \( x \) changes.
• It makes the plotting process on a coordinate system straightforward and systematic.

For instance, choosing \( x \) values from \(-3\) to \(3\) in the original solution helps illustrate how \( f(x) = x^2 - 4 \) manifests as a parabola. Observing how the values form a symmetric outline around the vertex is crucial too.

Vertex of Parabola

The vertex of the parabola is a key point and acts as the "turning point" of the curve. For the function \( f(x) = x^2 - 4 \), the vertex is at \((0, -4)\). This point is the lowest point of the parabola since it opens upwards.
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The vertex can be thought of as:

• The point where the parabola changes direction.
• A balance point of symmetry. Values of \( f(x) \) are mirrored on either side.
• For \( y = ax^2 + bx + c \), where there's no linear term \( (b = 0) \), the vertex lies on the y-axis.

In our example, the vertex \((0, -4)\) is also the minimum point of the function \( f(x) \). Recognizing the location and significance of the vertex helps in sketching an accurate graph of the quadratic function.