Problem 11 - Algebra and Trigonometry

Question

Find the coordinates of the vertex for the parabola defined by the given quadratic function. $$ f(x)=-2(x+1)^{2}+5 $$

Step-by-Step Solution

Answer

The vertex of the parabola defined by the function is \((-1, 5)\).

Step 1: Understand the Vertex Form of a Parabola

The vertex form of a parabolic function is \( f(x) = a(x-h)^{2}+k \) where \((h,k)\) is the vertex of the parabola.

Step 2: Identify the Values of \(h\) and \(k\)

From the given function \( f(x) = -2(x+1)^{2} + 5 \), we can see that \(h = -1\) and \(k = 5\). \(h\) is the opposite sign of the value inside the brackets and \(k\) is the constant

Step 3: Formulate the Vertex

From Step 2, we know that \(h = -1\) and \(k = 5\). These values form the coordinates of the vertex. Therefore, the vertex of the parabola given by the function is \((-1, 5)\).

Key Concepts

Vertex Form of a Parabola

The vertex form of a parabola is a special way to write the equation of a quadratic function that makes it easy to identify the vertex of the parabola. It is expressed as:\[f(x) = a(x-h)^2 + k\]Here are the important parts you should understand:

• \(a\) represents the vertical stretch or compression of the parabola. If \(a\) is positive, the parabola opens upwards. If it's negative, the parabola opens downwards.
• \((h, k)\) is the vertex of the parabola. \(h\) is the horizontal shift, and \(k\) is the vertical shift from the origin.

This form is particularly useful because it immediately tells you the vertex without additional calculations. When you convert a quadratic function from standard form to vertex form, you can quickly see how the graph will behave, where it will be "centered" on the x-axis, and how high or low it will go.
The vertex form is especially handy for graphing the function by hand, as it gives a clear starting point: the vertex.

Coordinates of the Vertex

The coordinates of the vertex of a parabola are essential as they give the highest or lowest point of the parabola, depending on whether it opens upwards or downwards. From the vertex form, we know that the vertex has coordinates \((h, k)\).
In the example of the quadratic function \(f(x) = -2(x+1)^{2} + 5\), identifying \(h\) and \(k\) is simple and direct:

• Look inside the parentheses: \((x+1)\). Here, \(x + 1 = x - (-1)\), which tells us \(h = -1\).
• The number added or subtracted outside the square is \(+5\), which means \(k = 5\).

Therefore, the vertex coordinates for this function are \((-1, 5)\). This point on the graph of the function shows where the parabola will have its maximum or minimum value, which is crucial for understanding the overall shape and position of the graph.

Parabola

A parabola is a U-shaped curve that is the graph of a quadratic function. In mathematical terms, a quadratic function is any function that can be written in the form \(f(x) = ax^2 + bx + c\).
Here are some important properties of parabolas:

• The direction that the parabola opens (upward or downward) is determined by the sign of the coefficient \(a\).
• The vertex represents the maximum or minimum point due to the symmetry of parabolas.
• A parabola is symmetric about its axis of symmetry, a vertical line that passes through the vertex.

The vertex form ties closely with these properties, as it makes it easier to draw and visualize the parabola's direction and position with respect to its vertex. By using the vertex coordinates, you can sketch the parabola, knowing it is going to open in the direction dictated by \(a\) and centered around the vertex \((h, k)\). This fundamental curve is not just theoretical; it applies to many real-world situations, including projectile motion and the design of reflective structures like satellite dishes.