Problem 56 - Algebra for College Students

Question

Give a step-by-step description of how you would go about graphing the parabola \(x^{2}-2 x-4 y-7=0\).

Step-by-Step Solution

Answer

Rewrite, complete the square, identify vertex, plot points, sketch curve.

Step 1: Rewrite Equation in Standard Form

The given equation is \(x^2 - 2x - 4y - 7 = 0\). To graph a parabola, we need to rewrite this in standard form for a parabola. The equation can be rewritten by organizing and isolating the \(y\) terms: \(x^2 - 2x - 7 = 4y\). Solving for \(y\), we get \(y = \frac{1}{4}(x^2 - 2x - 7)\). This equation is now in the form \(y = a(x - h)^2 + k\).

Step 2: Complete the Square

To further simplify \(y = \frac{1}{4}(x^2 - 2x - 7)\), we will complete the square for the \(x\)-terms. The expression \(x^2 - 2x\) can be transformed by adding and subtracting \(1\) (the square of half the coefficient of \(x\)): \((x - 1)^2 - 1\). Substituting back, we have: \(y = \frac{1}{4}((x - 1)^2 - 1 - 7)\). This simplifies to \(y = \frac{1}{4}(x - 1)^2 - 2\).

Step 3: Identify the Vertex and Direction of Opening

The standard form \(y = \frac{1}{4}(x - 1)^2 - 2\) indicates a parabola with vertex \((h, k) = (1, -2)\) and opens upwards since \(a = \frac{1}{4} > 0\). The coefficient \(\frac{1}{4}\) also determines the width of the parabola.

Step 4: Plot the Vertex

Start by plotting the vertex on the graph at the point \((1, -2)\).

Step 5: Determine and Plot Additional Points

To plot additional points, choose \(x\) values around the vertex and calculate their corresponding \(y\) values from the equation \(y = \frac{1}{4}(x - 1)^2 - 2\). For instance, if \(x = 0\), \(y = \frac{1}{4}(0 - 1)^2 - 2 = -\frac{9}{4}\), plot the point \((0, -\frac{9}{4})\). Do likewise for other values like \(x = 2\) to find \(y = -\frac{9}{4}\) and plot \((2, -\frac{9}{4})\).

Step 6: Sketch the Parabola

Using the vertex and the additional points, draw a smooth curve to depict the parabola. Ensure the curve opens upwards and is symmetric about the vertical line through the vertex \(x = 1\).

Key Concepts

Completing the Square

Completing the square is a helpful technique when working with quadratic expressions, especially when dealing with parabolas. This method makes it easier to convert a quadratic equation into vertex form.

• To complete the square, consider a quadratic expression of the form: \(x^2 + bx\).
• The goal is to transform this expression into a perfect square trinomial.
• You achieve this by adding and subtracting \((\frac{b}{2})^2\) within the expression.

This means if you start with: \(x^2 - 2x\), you calculate \((\frac{-2}{2})^2 = 1\). Hence, you rewrite \(x^2 - 2x\) as \((x-1)^2 - 1\).
By completing the square, you can then express your quadratic in a more convenient form for graphing, called the vertex form.

Standard Form of a Parabola

The standard form of a parabola is crucial for understanding the shape and position of the curve in the coordinate plane. A parabola can be expressed in the form:

• \(y = a(x-h)^2 + k\)

where \((h, k)\) is the vertex of the parabola, and \(a\) affects the direction and width of the opening. This form reveals several important features:

• The parabola opens upwards if \(a > 0\) and downwards if \(a < 0\).
• The vertex form makes it easy to graph since you can immediately see the vertex \((h,k)\).
• The value of \(a\) also influences the steepness or flatness of the parabola.

For example, the equation \(y = \frac{1}{4}(x-1)^2 - 2\) demonstrates this structure, showing a vertex at \((1, -2)\) and an opening upwards.

Vertex of a Parabola

The vertex of a parabola is a pivotal point that defines the curve's highest or lowest location. It's an easy-to-spot feature on a graph that indicates the maximum or minimum value of the quadratic function.

• In vertex form \(y = a(x-h)^2 + k\), the vertex is \((h, k)\).
• From this point, the parabola extends symmetrically.
• The vertex is also where the axis of symmetry intersects the parabola.

For the equation \(y = \frac{1}{4}(x-1)^2 - 2\), the vertex \((1, -2)\) is easy to locate. Placed at \(x = 1\), it's the point where the parabola makes its sharpest turn. Understanding this concept helps in quickly sketching or analyzing parabolas.

Plotting Points

Plotting points is a straightforward method to visualize the shape of the parabola. Once you have expressed the quadratic equation in a recognizable form, use this simple step to create an accurate graph.

• Begin by plotting the vertex, your key reference point.
• Choose several \(x\) values around this point.
• Calculate the corresponding \(y\) values using the equation.
• Plot these pairs of \((x, y)\) values.

For example, from the equation \(y = \frac{1}{4}(x-1)^2 - 2\), take \(x = 0\) and find that \(y = -\frac{9}{4}\). Plot \((0, -\frac{9}{4})\) and do similarly for other \(x\) values like \(2\), confirming the symmetry of the parabola around the vertex. Connect these points smoothly to display the complete parabola.