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 Parabola Details

The parabola has its vertex at the origin, which means the vertex is at the point \(0, 0\). Since the axis is along the \(x\)-axis, the parabola opens either to the right or left. We need the equation of the parabola in the standard form of \(y^2 = 4px\) or \(y^2 = -4px\).

Step 2: Use Point to Determine Equation

We need to use the given point \((-2, 4)\) to find the correct 'p' value. For the equation \(y^2 = 4px\), substitute \(y = 4\)and \(x = -2\), yielding the equation \(16 = 4p(-2)\)which simplifies to \(16 = -8p\). Solve for \(p\) to get \(p = -2\). So, the equation is \(y^2 = -8x\).

Step 3: Draw the Parabola

For the sketch, plot the vertex at \(0,0\). Since the parabola opens to the left due to the negative sign in \(y^2 = -8x\), it moves towards negative \(x\)-values. The point \((-2, 4)\) lies on this curve, confirming the shape and direction.

Key Concepts

Vertex Form

The vertex form is a helpful way to write the equation of a parabola. When we know the vertex's position, it becomes easier to understand the parabola's shape. The standard vertex form is given by:

• \( y = a(x - h)^2 + k \)

Here, \((h, k)\) is the vertex of the parabola. However, in the case where the vertex is at the origin, \((0, 0)\), this simplifies to

• \( y = ax^2 \)
• or \( x = ay^2 \)

The second form is particularly useful when the parabola opens left or right along the x-axis. By substituting known points into these forms, you can find ‘a’ or the equivalent ‘p’ in equations where it opens sideways. This further simplifies forming the equation of your parabola, especially when given specific points it passes through.

Conic Sections

Parabolas are special because they are one of the four types of conic sections. Conic sections are shapes that are formed by slicing a double cone with a plane. The four sections include:

• Circle
• Ellipse
• Parabola
• Hyperbola

A parabola results from a plane cutting parallel to the edge of the cone. Understanding conic sections helps contextually when looking at parabolas, especially their properties and orientation. Parabolas have specific focuses and directrices, and these contribute to equations like \( y^2 = 4px \) which describe parabolas effectively. Through these, you can determine properties such as direction and width.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is all about plotting mathematical equations into graphs. This branch of mathematics uses numbers to describe geometrical shapes. For parabolas, coordinate geometry allows us to find equations based on given points and vertex positions using coordinates. In the given problem, the parabola goes through the point ((-2,4)) with the vertex at the origin (0,0). By applying coordinate geometry, you substitute these coordinates into the general parabola equation. This helps find the best-fit equation. Coordinate geometry simplifies visualizing algebraic expressions, thus making it easy to sketch graphs, determine points of intersection, or find distances between points and shapes.