Problem 14 - Algebra 2

Question

Graph each function. $$ y=-(x-5)^{2}-3 $$

Step-by-Step Solution

Answer

The parabola opens downwards with vertex at (5, -3).

Step 1: Identify the Vertex Form

The given function is in the vertex form of a quadratic equation, which is: \[ y = a(x-h)^2 + k \]Here, the vertex is at the point \((h, k)\). In our function, \(a = -1\), \(h = 5\), and \(k = -3\), meaning the vertex is \((5, -3)\).

Step 2: Determine the Direction of the Parabola

Since \(a = -1\) (a negative value), the parabola opens downwards. This means the graph will be a downward-facing U-shape, or an upside-down parabola.

Step 3: Identify Additional Points

To accurately graph the function, identify additional points by choosing several values for \(x\) around the vertex and calculating the corresponding \(y\) values. For example, if \(x = 4\), then:\[ y = -(4 - 5)^2 - 3 = -(1)^2 - 3 = -1 - 3 = -4 \]Thus, one point on the parabola is \((4, -4)\). You can similarly find points for \(x = 6\), \(x = 3\), and \(x = 7\).

Step 4: Sketch the Graph

Plot the vertex \((5, -3)\) and the additional points you calculated, such as \((4, -4)\) and \((6, -4)\). Draw a smooth curve connecting these points, ensuring it opens downwards, reflecting the shape of the parabola as described.

Key Concepts

Vertex Form of a Quadratic

In the world of quadratic functions, the vertex form is a particularly handy format for graphing and understanding parabolas. The vertex form of a quadratic equation is expressed as \( y = a(x-h)^2 + k \).

Here, \(a\), \(h\), and \(k\) play significant roles in shaping the parabola:

• \(a\): This coefficient affects the width and direction of the parabola.
• \(h\): The x-coordinate of the vertex.
• \(k\): The y-coordinate of the vertex.

For the function \( y = -(x-5)^2 - 3 \), the values are \(a = -1\), \(h = 5\), and \(k = -3\).

This gives a vertex of \((5, -3)\). Understanding the vertex form allows you to easily identify the vertex and make informed guesses about the general shape and position of the parabola on a graph.

Parabola Direction

The direction in which a parabola opens depends on the sign of the coefficient \(a\) in the vertex form \( y = a(x-h)^2 + k \). If \(a\) is positive, the parabola opens upwards, forming a U-shape. However, if \(a\) is negative, as in our function \( y = -(x-5)^2 - 3 \) where \(a = -1\), the parabola opens downwards.

This creates a downward-facing U-shape, sometimes likened to an upside-down umbrella. The downward direction indicates that the vertex represents the highest point on the graph, making it crucial for determining the parabola's range of values.

Vertex Identification

Recognizing and locating the vertex on a graph is essential since it represents the extremum point of the parabola. The vertex in a quadratic equation in vertex form \( y = a(x-h)^2 + k \) is directly given by the coordinates \((h, k)\).

In the function \( y = -(x-5)^2 - 3 \), the vertex is at \((5, -3)\).

Identifying the vertex provides a fixed point around which the rest of the parabola is symmetrically arranged.

This makes it a key tool in sketching the graph accurately and understanding the transformation from a standard quadratic function to its actual graph on a coordinate plane.

Plotting Points on a Graph

To sketch a complete parabola, you need more than just the vertex. Plotting additional points helps in shaping the whole parabola accurately. Choose x-values close to the vertex, then substitute them into the equation to find corresponding y-values.

For \(y = -(x-5)^2 - 3\):

• When \( x = 4 \), \( y = -(4-5)^2 - 3 = -4 \), giving the point \((4, -4)\).
• Similarly, for \( x = 6 \), \( y = -(6-5)^2 - 3 = -4 \), yielding \((6, -4)\).

Plot these points on the graph, and draw a smooth curve through them and the vertex \((5, -3)\). By doing so, you ensure the parabola reflects the proper U-shape and opening direction indicated by the value of \(a\). This step solidifies the accuracy of your graph and enhances your understanding of the parabola's geometric properties.