Problem 84 - Algebra 2

Question

Graph each function. $$ y=-2(x+1)^{2}-3 $$

Step-by-Step Solution

Answer

The parabola has its vertex at \((-1, -3)\), axis of symmetry \(x = -1\), and opens downwards.

Step 1: Identify the Vertex

The given function is in vertex form, which is \(y = a(x-h)^2 + k\), where \((h,k)\) is the vertex of the parabola. For the given function \(y=-2(x+1)^2-3\), the vertex is \((-1, -3)\).

Step 2: Determine the Axis of Symmetry

The axis of symmetry for a parabola in vertex form is \(x = h\). Therefore, the axis of symmetry for the function is \(x = -1\).

Step 3: Find the Direction of the Parabola

The coefficient of \((x+1)^2\) is \(-2\). Since it is negative, the parabola opens downwards.

Step 4: Plot the Vertex

Start by plotting the vertex \((-1, -3)\) on the coordinate plane.

Step 5: Plot Additional Points and Draw the Parabola

To graph the parabola, choose points on either side of the axis of symmetry and calculate their corresponding \(y\) values to plot them. Use your symmetry about the axis of symmetry \(x = -1\) to find points on the other side of the vertex. Then sketch the parabola through these points, making sure it opens downwards.

Key Concepts

Vertex Form of a Parabola

Understanding the vertex form of a parabola is fundamentally important for graphing it accurately. The general equation is given by (y = a(x-h)^2 + k), where (h, k) represents the vertex, a determines the parabola's width and its direction (up or down).

When graphing the function (y=-2(x+1)^2-3), identifying the vertex is the first step. Here, we have h as -1 and k as -3, revealing the vertex's coordinates as (-1, -3). The vertex represents the highest or lowest point on the graph, serving as a pivotal reference for subsequent steps.

In our exercise, we recognize the minus sign in front of the 2 as indicative of the reflective property on the x-axis, suggesting our graph will open downward. Additionally, the 2 indicates the parabola will be narrower than the standard y=x^2, due to it being greater than 1 in absolute value.

Axis of Symmetry

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. For a parabola in vertex form, this axis is always x = h, where h is the x-coordinate of the vertex.

Using the equation y=-2(x+1)^2-3, we identify the axis of symmetry to be x = -1. This implies that every point on the parabola has a correspondent on the opposite side of this line at an equal distance. The axis of symmetry informs us about the nature of the parabola's curve and helps ensure our graph is well-proportioned and accurate.

When plotting points for the graph, the axis of symmetry enables us to only calculate the y-value for certain x-coordinates. We can reflect these across the axis of symmetry to obtain additional points needed to draw our parabola, streamlining the graphing process significantly.

Direction of a Parabola

The direction of a parabola refers to whether it opens upwards or downwards and is essential for understanding the overall shape of the graph. This aspect is governed by the coefficient a in the vertex form equation y = a(x-h)^2 + k.

If a is positive, the parabola opens upwards, resembling a 'U' shape. Contrarily, if a is negative, as in our example y = -2(x+1)^2 - 3, the parabola opens downwards, creating an 'n' shape. This downward direction indicates that the vertex is the highest point on the graph. As the absolute value of a increases, the parabola becomes narrower. Conversely, as it approaches zero, the parabola widens.

Knowing the direction helps us understand the parabola's behavior relative to its vertex. It guides us in sketching the graph's curvature and in anticipating the range of the function's values which, in our downward-opening example, have a maximum at the vertex and decrease as we move away from it.