Problem 18 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the equation of the parabola whose vertex is the origin and whose axis is the \(y\)-axis if the parabola passes through the point \((-3,5)\). Make a sketch.

Step-by-Step Solution

Answer

The parabola is \( y = \frac{5}{9}x^2 \).

Step 1: Understanding the Vertex Form Equation

Since the vertex is at the origin and the axis is the y-axis, the equation of the parabola in vertex form is \( y = ax^2 \). This equation represents a parabola that opens upwards or downwards depending on the value of \(a\).

Step 2: Substitute the Given Point into the Equation

The parabola passes through the point \((-3, 5)\). Substitute \(x = -3\) and \(y = 5\) into the equation \(y = ax^2\): \[ 5 = a(-3)^2 \].

Step 3: Solving for the Coefficient a

Simplify the equation to find \(a\): \[ 5 = 9a \].Divide both sides by 9: \( a = \frac{5}{9} \).

Step 4: Write the Equation of the Parabola

Now that we have found the value of \(a\), we can write the equation of the parabola as:\[ y = \frac{5}{9}x^2 \].

Step 5: Sketching the Parabola

To sketch the parabola, note that it opens upwards since \(a > 0\). Its vertex is at the origin, \((0,0)\), and it passes through the point \((-3,5)\). It is symmetric about the y-axis.

Key Concepts

Vertex Form of a Parabola

The vertex form of a parabola is a specific way of writing the quadratic equation that emphasizes the vertex's position. When the vertex of the parabola is at the origin and the axis of symmetry is aligned with the y-axis, the vertex form simplifies to:

• \( y = ax^2 \)

This shows that if the vertex is at the origin \(0,0\) and the parabola opens upwards or downwards, the equation will take this basic shape.
The variable \(a\) in the equation \(y = ax^2\) controls the parabola's direction and "width".

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), the parabola opens downwards.

Additionally, the absolute value of \(a\) determines how "tight" or "wide" the parabola appears. Larger values result in a steeper and narrower parabola, while smaller values lead to a broader curve.

Y-Axis Symmetry in Parabolas

Parabolas like the one in this exercise can exhibit symmetry about the y-axis. This symmetry means that if you were to fold the graph of the parabola along the y-axis, both halves would match perfectly.

• This is because, for any point \((x, y)\) on the parabola, there will be a corresponding point \((-x, y)\) that lies directly opposite across the y-axis.

Understanding this symmetry is useful when graphing the parabola, as it helps to predict the shape of the curve on both sides of the axis without needing to calculate each point. This type of symmetry occurs specifically with vertical parabolas, which have their vertex either at or directly along the y-axis.

Sketching Parabolas: Practical Tips

Sketching parabolas involves plotting a few key points and understanding the general shape dictated by the parabola's equation. Here's a simple approach:Identify the Vertex:

• For the parabola \( y = \frac{5}{9}x^2 \), the vertex is at the origin, \((0,0)\).

Determine the Direction:

• Since \( a = \frac{5}{9} \) and \(a > 0\), the parabola opens upwards.

Plot Additional Points:

• Use symmetry and the equation to find other key points. For instance, if it passes through \((-3, 5)\), it will also pass through \((3, 5)\) due to symmetry.

Draw the Curve:

• Start at the vertex, then draw a smooth curve through the plotted points, ensuring it's symmetrical about the y-axis.

By understanding these elements, sketching a parabola becomes systematic and straightforward, balancing precision with approximate visual representation.