Problem 45 - Contemporary Precalculus

Question

In Exercises \(43-54\), find the equation of the parabola satisfying the given conditions. Vertex (1,0)\(;\) axis \(x=1 ;(2,13)\) on graph.

Step-by-Step Solution

Answer

Answer: The equation of the parabola is \((x-1)^2=\frac{1}{13}(y)\).

Step 1: Identify the Vertex and Axis of the Parabola

The given vertex (h,k) is (1,0), and since the axis is vertical (x=1), we know that the parabola has an equation of the form \((x-h)^2=4a(y-k)\) where a is a constant that determines the direction and stretch of the parabola.

Step 2: Determine the Direction of the Parabola

We have a point (2,13) on the parabola, we can use this information to find out if the parabola opens upwards or downwards. The vertex is (1,0) which has a smaller x coordinate than the point (2,13). Since the parabola is symmetric along its axis, half the parabola (the half containing the point (2,13)) must open upwards. Hence the parabola indeed opens upwards.

Step 3: Find the Value of a

Now let's substitute the vertex (h,k)=(1,0) and the point (2,13) in the equation of the parabola to find the value of a: \((x-h)^2=4a(y-k)\) and substitute the point (2,13), \((2-1)^2=4a(13-0) \Rightarrow 1=52a\) Solve for a: \(a=\frac{1}{52}\)

Step 4: Write the Equation of the Parabola

Now that we have found the values of h, k, and a, we can write the equation of the parabola: \((x-h)^2=4a(y-k)\) Substitute the values of h, k, and a to get: \((x-1)^2=4\left(\frac{1}{52}\right)(y-0)\) Simplify to get the final equation of the parabola: \((x-1)^2=\frac{1}{13}(y)\)

Key Concepts

Vertex Form

The vertex form of a parabola's equation is an incredibly useful way to understand the graph's features at a glance. In this form, the parabola is expressed as \[(x-h)^2 = 4a(y-k)\]. This equation is helpful because it immediately gives us the vertex of the parabola at the point \((h, k)\).

The vertex represents the highest or lowest point on the graph, depending on whether the parabola opens upwards or downwards. It is also the point along the axis of symmetry, making it a crucial component for sketching the graph correctly. Having the vertex form makes it easy to convert between different forms of quadratic expressions and to understand the geometric significance of each component.

Symmetry of Parabolas

Parabolas are symmetrical along a specific line called the axis of symmetry. This line passes through the vertex, effectively dividing the parabola into two mirror-image halves. For a parabola in the form \((x-h)^2 = 4a(y-k)\), the symmetry axis is vertical along \(x = h\).

This symmetry means that for any point on one side of the parabola, there is a corresponding point on the opposite side at the same height. When dealing with equations and graphs, remembering this characteristic helps in confirming calculations and predicted values.

In problems involving symmetry, like identifying whether a parabola opens upwards or downwards, the symmetry property helps easily predict where another point might lie based on its position relative to the axis.

Parabola Direction

Determining the direction in which a parabola opens involves understanding the role of the constant \(a\) in the vertex form equation. For the equation \((x-h)^2 = 4a(y-k)\), the direction of opening is directly linked to the sign of \(a\).

- If \(a > 0\), the parabola opens upwards; it resembles a "U" shape.- If \(a < 0\), the parabola opens downwards, similar to an upside-down "U".

By understanding the vertex and another given point, students can substitute into the equation to discover \(a\), thereby confirming the direction. In our case, the point \((2,13)\) helped us deduce that with \(a = \frac{1}{52}\), the parabola opens upwards.

Finding a in Parabolas

To find the constant \(a\) in the vertex form equation, which affects the parabola's direction and shape, you need at least one additional point on the graph besides the vertex. This point is substituted into the vertex form equation to solve for \(a\).

Using the given example, we have the vertex at \((1,0)\) and a point \((2,13)\). Substituting these into the vertex form:\[(x-1)^2 = 4a(y-0)\], we simplify with \(x = 2\) and \(y = 13\) to find \(a\):
1. Substitute: \((2-1)^2 = 4a(13-0)\)2. Solve: \(1 = 52a\) 3. \(a = \frac{1}{52}\)

This value of \(a\) gives insights into the parabola's stretch, with larger values leading to narrower parabolas.