Problem 53 - Algebra 2

Question

Graph each function. $$ y=\frac{1}{3}(x+5)^{2}-1 $$

Step-by-Step Solution

Answer

Graph the parabola with vertex at (-5, -1), axis of symmetry x = -5, and opening upwards.

Step 1: Identify the Components of the Function

The given function is in the form of a quadratic equation in vertex form, which is \( y = a(x - h)^2 + k \). Here, \( a = \frac{1}{3} \), \( h = -5 \), and \( k = -1 \). This tells us that the parabola has a vertex at \((-5, -1)\) and is opened upwards because \( a \) is positive.

Step 2: Determine the Axis of Symmetry

The axis of symmetry for a parabola in vertex form \( y = a(x-h)^2 + k \) is \( x = h \). For this function, the axis of symmetry is \( x = -5 \). This is a vertical line that passes through the vertex and divides the parabola into mirror images.

Step 3: Identify the Vertex

The vertex is a critical point of the parabola and is located at \( (h, k) = (-5, -1) \). This point is the minimum point of the parabola because the parabola opens upwards.

Step 4: Determine Additional Points

To graph the function accurately, choose x-values around the vertex such as \(-4\), \(-3\), etc., and compute their corresponding y-values. For example, when \( x = -4 \), \( y = \frac{1}{3}((-4)+5)^2 - 1 = rac{1}{3}(1) - 1 = -0.6667 \).

Step 5: Sketch the Graph

Plot the vertex at \((-5, -1)\) on the graph. Draw the axis of symmetry as a dashed vertical line through \( x = -5 \). Plot the additional points calculated in Step 4 on the graph. Use the axis of symmetry to reflect these points for more symmetry. Connect these points with a smooth curve to form the parabola.

Key Concepts

Vertex Form

The vertex form of a quadratic equation is essential for easily identifying the properties of a parabola. It is expressed as \( y = a(x - h)^2 + k \). This format clearly shows the vertex of the parabola, which is the point \((h, k)\). In this form, the coefficient \( a \) influences the shape and direction of the parabola. If \( a \) is positive, the parabola opens upwards, and if negative, it opens downwards. The absolute value of \( a \) affects the parabola's width:

• A larger \( |a| \) value makes it steeper.
• A smaller \( |a| \) value makes it wider.

For the function \( y = \frac{1}{3}(x + 5)^2 - 1 \), the vertex form tells us the vertex is at \((-5, -1)\), giving us a starting point to graph the function.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola, serving as a mirror line that divides the parabola into two symmetrical halves. For a quadratic equation in vertex form, \( y = a(x - h)^2 + k \), the axis of symmetry is simply \( x = h \). This is because the parabola is symmetrical around this line. In the example function, \( x = -5 \) is the axis of symmetry. By knowing this, we can draw a dashed vertical line at \( x = -5 \) on the graph, providing a visual guide to help plot the points and ensure that the parabola is accurately sketched.

Parabola

A parabola is the graph of a quadratic function and is characterized by a U-shape that either opens upwards or downwards. The vertex of the parabola is a focal point, such as the highest or lowest point of the graph, depending on the direction in which it opens. In our example, since \( a = \frac{1}{3} \) and is positive, the parabola opens upwards. This indicates that the vertex at \((-5, -1)\) represents the minimum point of the graph. Understanding the properties of a parabola helps in predicting the direction of its curve and its behavior as it extends away from the vertex.

Graphing Functions

Graphing quadratic functions involves several steps that ensure accuracy and a clear understanding of its shape. Once the vertex and axis of symmetry are identified, additional points are crucial:

• Select x-values around the vertex.
• Calculate the corresponding y-values using the function equation.

Plot these points on a coordinate plane to visualize the curve.
Reflect these points across the axis of symmetry for any additional symmetry points.
Finally, connect all the plotted points with a smooth, curved line to complete the graph of the parabola. Properly graphing functions provides a clear illustration of how the function behaves, making it easier to analyze and understand.