Problem 30 - Algebra A Combined Function

Question

Graph each quadratic function. Label the vertex and sketch and label the axis of symmetry. $$ g(x)=4(x-4)^{2}+2 $$

Step-by-Step Solution

Answer

The vertex is \((4, 2)\) and the axis of symmetry is \(x = 4\).

Step 1: Identify the Vertex

The given quadratic function is in vertex form, which is given as \( g(x) = a(x-h)^2 + k \). Comparing with the function \( g(x) = 4(x-4)^2 + 2 \), we identify that \( h = 4 \) and \( k = 2 \). Thus, the vertex is \((4, 2)\).

Step 2: Determine the Axis of Symmetry

The axis of symmetry for a parabola in the form \( g(x) = a(x-h)^2 + k \) is the vertical line \( x = h \). Therefore, for our function, the axis of symmetry is \( x = 4 \).

Step 3: Sketch the Parabola

Since the coefficient \( a = 4 \) is positive, the parabola opens upwards. Using the vertex \((4, 2)\) and the axis of symmetry \( x = 4 \) as a guide, sketch an upward-opening parabola centered around the axis of symmetry.

Step 4: Label the Vertex and Axis of Symmetry on the Graph

Clearly mark the vertex at \((4, 2)\) on your sketch. Draw and label the axis of symmetry \( x = 4 \), ensuring it is a dashed line to indicate it's an axis rather than part of the graph itself.

Key Concepts

Vertex Form

In quadratic functions, the vertex form is a way of expressing parabolas to easily identify their key features. It looks like this: \( g(x) = a(x-h)^2 + k \). Here, \( (h, k) \) represents the coordinates of the vertex, the peak or trough of the parabola. The parameter \( a \) indicates the direction and the "width" of the parabola.
The vertex form is particularly useful because you can quickly determine the vertex just by looking at the equation. In our example, \( g(x) = 4(x-4)^2 + 2 \), the vertex form tells us that the vertex is at \( (4, 2) \). This provides a central point around which the parabola curves.

Axis of Symmetry

The axis of symmetry in a parabola is a vertical line that runs through the vertex. It divides the parabola into two mirror-image halves. For a quadratic function in vertex form \( g(x) = a(x-h)^2 + k \), the axis of symmetry is given by the equation \( x = h \).
In simpler terms, this line helps us understand that for any two points on either side of a parabola that are equidistant from this axis, their y-values will be the same. For the function \( g(x) = 4(x-4)^2 + 2 \), the axis of symmetry is \( x = 4 \), providing a guide for how we sketch the parabola.

Graphing Parabolas

Graphing parabolas involves understanding how the equation is translated into a shape on the graph. Begin by plotting the vertex on your coordinate system. Here, the vertex \( (4, 2) \) is the starting point. Then, draw the axis of symmetry, \( x = 4 \), as a dashed line to remind that it's not part of the graph but a guideline.
A positive coefficient \( a \) in the equation, like \( a = 4 \), indicates the parabola opens upwards. Plot several points on either side of the vertex to see how the parabola extends, using the axis of symmetry to ensure symmetry. Connect these points in a smooth U-shaped curve extending from the vertex.

Upward-Opening Parabola

An upward-opening parabola appears as a "U" shape on the graph. The direction in which the parabola opens depends on the sign of the coefficient \( a \) in the quadratic equation. If \( a \) is positive as in \( g(x) = 4(x-4)^2 + 2 \), the parabola opens upwards. This means that the further you move away from the vertex, the y-values of the parabola increase.
In quadratic equations, this shape signifies minimum points since the vertex is the lowest point before the parabola curves upwards. It's important to understand this concept when analyzing or sketching graphs in algebra or more advanced mathematics.