Problem 46 - Precalculus: Mathematics for Calculus

Question

Find an equation for the conic section with the given properties. The parabola that passes through the point \((6,-2),\) with vertex \(V(4,-1)\) and vertical axis of symmetry

Step-by-Step Solution

Answer

The equation is \( y = -\frac{1}{4}(x-4)^2 - 1 \).

Step 1: Identify vertex form of a parabola

The equation of a parabola with a vertical axis of symmetry and vertex \( (h, k) \) is \( y = a(x-h)^2 + k \). For this parabola, the vertex \( V \) is \( (4, -1) \), so the equation becomes \( y = a(x-4)^2 - 1 \).

Step 2: Substitute given point into the equation

To find the value of \( a \), substitute the coordinates of the point \((6, -2)\) into the equation: \( -2 = a(6-4)^2 - 1 \).

Step 3: Solve for the coefficient \( a \)

Simplify the equation: \( -2 = a(2)^2 - 1 \) \( -2 = 4a - 1 \)\( 4a = -1 \)\( a = -\frac{1}{4} \).

Step 4: Write the equation of the parabola

Substitute \( a = -\frac{1}{4} \) back into the vertex form equation: \( y = -\frac{1}{4}(x-4)^2 - 1 \). This is the equation of the parabola.

Key Concepts

Vertex Form

The vertex form is one of the ways to express the equation of a parabola. It's particularly useful because it gives you clear information about the parabola's vertex, which is the highest or lowest point depending on the parabola's orientation. This form is written as:

• Equation: \( y = a(x-h)^2 + k \)
• \((h, k)\) represents the vertex of the parabola.
• The variable \( a \) determines the direction and width of the parabola.

Knowing the vertex form makes it easier to graph a parabola on the coordinate plane because you can quickly see the vertex's location and how the parabola opens.

Axis of Symmetry

The axis of symmetry is a line that divides the parabola into two mirror-image halves. For a parabola with a vertical axis like the one in our exercise, the axis of symmetry is a vertical line that passes through the vertex. In the vertex form equation, the axis of symmetry can be quickly identified using the following ideas:

• The equation of the axis of symmetry is \( x = h \), where \( h \) is the x-coordinate of the vertex.
• For our vertex \( (4, -1) \), the axis of symmetry would be the line \( x = 4 \).

This line helps in understanding the orientation of the parabola and is fundamental in sketching the graph of the quadratic function.

Vertex

The vertex is a crucial point on the parabola, acting as a turning point from which the parabola changes direction. It can be identified directly from the vertex form equation:

• In the equation \( y = a(x-h)^2 + k \), the vertex is \((h, k)\).
• A vertex \((4, -1)\) means that this is where the parabola either reaches its highest or lowest point.
• Since our parabola opens downwards (because \( a = -\frac{1}{4} \)), the vertex is a maximum point.

Understanding the vertex allows us to describe the general shape and position of the parabola on a graph.

Coefficient

The coefficient \( a \) in the vertex form equation \( y = a(x-h)^2 + k \) plays an essential role in determining a parabola's properties. It affects both the direction and the shape of the graph:

• When \( a > 0 \), the parabola opens upwards, resembling a U-shape.
• When \( a < 0 \), like in our exercise where \( a = -\frac{1}{4} \), the parabola opens downwards, resembling an upside-down U.
• The absolute value of \( a \) controls the width of the parabola. The smaller \(|a|\), the wider the parabola; the larger \(|a|\), the narrower the parabola.

By solving for \( a \), especially using given points, you can grasp the exact nature and direction of the parabola, which is crucial for accurate graphing and analysis of quadratic functions.