Problem 34 - Algebra A Combined Function

Question

Graph each quadratic function. Label the vertex and sketch and label the axis of symmetry. $$ G(x)=\frac{1}{5}(x+4)^{2}+3 $$

Step-by-Step Solution

Answer

Vertex: (-4, 3); Axis of Symmetry: x = -4.

Step 1: Identify the Vertex Form of the Function

The given function is already in vertex form which is \(G(x) = \frac{1}{5}(x+4)^2 + 3\). In vertex form, a quadratic is \(G(x) = a(x-h)^2 + k\). Comparing with \(a(x-h)^2 + k\), the vertex \((h, k)\) for this function is \((-4, 3)\).

Step 2: Determine the Axis of Symmetry

In a quadratic function in vertex form, \(x = h\) is the equation of the axis of symmetry. For this function, since \(h = -4\), the axis of symmetry is \(x = -4\).

Step 3: Sketch the Graph

Plot the vertex \((-4, 3)\) on a graph. Since \(a = \frac{1}{5}\) is positive and less than 1, the parabola opens upwards and is wider than the standard \(x^2\) parabola. Draw the axis of symmetry at \(x = -4\) as a vertical dashed line. Sketch the parabola using the vertex as a guide, ensuring it is symmetrical around the axis of symmetry.

Key Concepts

Vertex Form

In mathematics, a quadratic function can be expressed in something called the vertex form. The idea behind the vertex form is to provide an easily recognizable representation where you can directly identify key features of the parabola, like the vertex. The vertex form of a quadratic equation is written as \[ G(x) = a(x-h)^2 + k \]where:

• \( a \) is a coefficient that affects how wide or narrow the parabola is and whether it opens upwards or downwards. If \( a \) is positive, the parabola opens upwards; if negative, downwards.
• \( h \) and \( k \) are the coordinates of the vertex of the parabola, represented as \((h, k)\).

In the exercise, comparing \(G(x) = \frac{1}{5}(x+4)^2 + 3\) to the standard form, \( (-4, 3) \) is the vertex. Being in vertex form allows you to easily plot the graph by focusing on the vertex and understanding how "stretchy" or "compressed" the curve is based on the value of \( a \). This straightforward approach saves time and simplifies the process of graphing these mathematical entities.

Parabola

A parabola is a U-shaped curve that is the graphical representation of a quadratic function. Every quadratic function, no matter how complex, will graph into a parabola. The unique features of a parabola include its vertex, its direction (upward or downward), and its symmetry.

• The opening direction of the parabola is determined by the sign of the coefficient \( a \). In this exercise, \( a = \frac{1}{5} \), a positive value, means the parabola opens upwards.
• The vertex of the parabola is its highest or lowest point, depending on the direction it opens.
• The coefficient \( a \), which is less than 1, indicates the parabola is wider than the basic \( x^2 \) parabola graph, meaning it opens in a broader U-shape.

Understanding these characteristics helps in sketching the curve accurately. The parabola serves as the foundation for understanding quadratic functions and can be observed in various phenomena, from trajectories of projectiles to satellite dishes.

Axis of Symmetry

The axis of symmetry in a quadratic function is a crucial feature that runs vertically through the vertex of the parabola. This axis is an imaginary line that divides the parabola into two mirror-image halves, ensuring that for any point on one side of the parabola, there is a corresponding point on the opposite side at the same distance from the axis. To find the equation of the axis of symmetry when given the vertex form of a quadratic, you use the formula:\[ x = h \]where \( h \) comes directly from the vertex coordinates \((h, k)\). In our exercise, the vertex was identified as \((-4, 3)\), thus making the axis of symmetry \(x = -4\).Visualizing this axis is important when graphing because it acts as a guide for ensuring the parabola is evenly distributed on both sides, providing graphical balance to the function representation. When sketching on a graph, you might draw this line using a different color or a dashed line to emphasize its guiding role.