Problem 16 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the equation of the parabola through the point \((-2,4)\) if its vertex is at the origin and its axis is along the \(x\) -axis. Make a sketch.

Step-by-Step Solution

Answer

The equation of the parabola is \( y^2 = -8x \).

Step 1: Understand the Vertex Form of a Parabola

For a parabola with its vertex at the origin and its axis along the x-axis, its general equation is in the form \( y^2 = 4ax \), where \( a \) determines the distance of the focus of the parabola from the vertex. The equation will open in the direction of the x-axis (to the right if \( a > 0 \) and to the left if \( a < 0 \)).

Step 2: Substitute the Given Point into the Parabola Equation

The given point that lies on the parabola is \((-2, 4)\). Substitute \(x = -2\) and \(y = 4\) into the equation \( y^2 = 4ax \) to find the value of \( a \):\[ (4)^2 = 4a(-2) \] \[ 16 = -8a \]

Step 3: Solve for \( a \)

Now solve the equation \( 16 = -8a \) to find the value of \( a \):\[ a = \frac{16}{-8} \] \[ a = -2 \]

Step 4: Write the Final Equation of the Parabola

Now that we have determined \( a = -2 \), substitute this value back into the equation of the parabola. Thus, the parabola's equation is:\[ y^2 = -8x \]

Step 5: Sketch the Parabola

Since the parabola is in the form \( y^2 = kx \) with \( k < 0 \), it opens to the left. Plot the vertex at the origin \((0,0)\), and include the point \((-2, 4)\) which lies on the parabola. Draw a parabola opening to the left through this point.

Key Concepts

Vertex Form of a Parabola

The vertex form of a parabola is a crucial concept in understanding the properties of parabolic shapes. It is particularly useful because it allows us to easily identify the vertex of the parabola and how it behaves. For a parabola with its vertex at the origin, the equation can be either:

• \(x^2 = 4ay\) if it opens up or down, where the axis of symmetry is vertical
• \(y^2 = 4ax\) if it opens left or right, where the axis of symmetry is horizontal

The equation \(y^2 = 4ax\) tells us that the parabola's vertex is at \((0, 0)\), which is the simplest form for analyzing its properties. The variable \(a\) directly influences the width and direction of the parabola. A positive \(a\) signals the parabola opens to the right, while a negative \(a\) means it opens to the left. This form of the equation simplifies finding where the parabola intersects the x and y axes based on given points or conditions.

Focus of Parabola

The focus of a parabola is a point from which distances to the parabola reflect symmetry and focal properties. This focus is a key part of what makes a parabola unique. For parabolas in vertex form where the vertex is at the origin, the focus can be found by looking at the equation:

• In \(y^2 = 4ax\), the focus is at point \((a, 0)\) if the parabola opens to the right or left.

For the given example, we found that \(a = -2\), thereby determining the focus to be at \((-2, 0)\) for our parabola opening to the left. The role of the focus is crucial, as parabolas are defined such that any point on the parabola maintains an equal distance to the focus and the corresponding directrix line. Knowing the location of the focus helps in graphically representing the parabola and understanding its reflective properties, which are utilized in many real-world applications, such as satellite dishes and car headlights.

Opening Direction of Parabola

The opening direction of a parabola describes which way the "arms" of the parabola extend from the vertex. Given the equation \(y^2 = 4ax\), this direction is determined by the sign of \(a\). The parabola opens:

• To the right if \(a > 0\)
• To the left if \(a < 0\)

In our solved exercise, the determined value \(a = -2\) led us to conclude that the parabola opens to the left. Visualizing this helps in sketching the curve accurately. Since the parabola opens to the left, any point on it will appear closer to the left as it gets further from the vertex. Understanding the opening direction is especially beneficial in predicting the path the parabola will take when plotting, ensuring a correct and meaningful representation of the equation's graph. This directional understanding supports the process of setting up coordinate geometry situations and solving related problems efficiently.