Problem 22 - Algebra and Trigonometry

Question

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ f(x)=x^{2}+5 $$

Step-by-Step Solution

Answer

Shift the parabola \( x^2 \) up by 5 units.

Step 1: Identify the Base Function

The function given is \( f(x) = x^2 + 5 \). The base function here is \( g(x) = x^2 \), which is a standard quadratic function that has its vertex at the origin (0,0) and opens upwards.

Step 2: Understand the Transformation

The function \( f(x) = x^2 + 5 \) is obtained by adding 5 to the base function \( g(x) = x^2 \). This translates the graph of \( g(x) \) vertically upward by 5 units. The vertex of the parabola shifts from (0,0) to (0,5). The shape of the parabola remains the same.

Step 3: Sketch the Transformed Graph

Start by sketching the base graph \( g(x) = x^2 \), which is a parabola with its vertex at the origin and symmetric about the y-axis. Next, shift this entire graph upward by 5 units to account for the transformation. The new vertex will be at (0,5). The graph is still symmetric around the y-axis and opens upwards with the same curvature.

Key Concepts

Graph Transformations

Graph transformations allow us to modify and shift the appearance of basic graphs such as those of quadratic functions. These transformations include shifting, stretching, compressing, and reflecting the graph across the axes. When dealing with a quadratic function like \( f(x) = x^2 + 5 \), we often refer to vertical transformations. These involve moving the entire graph either upward or downward without altering its shape.

• The addition or subtraction within the function (like the "+5" in \( f(x) = x^2 + 5 \)) indicates a vertical transformation.
• In this example, the graph is moved upwards by 5 units.
• The vertical shift does not change the symmetry or the direction in which the parabola opens, it simply moves it to a new position on the coordinate plane.

Understanding graph transformations is key to efficiently sketching graphs without having to plot multiple points.

Function Translation

Function translation is a specific type of transformation that shifts the entire graph of a function along the coordinate plane. In our example, the function \( f(x) = x^2 + 5 \) is a translated version of the base function \( g(x) = x^2 \). Function translation involves moving the graph:

• Vertically - by adding or subtracting a constant value from the function, such as the "+5" in this case, which moves the graph upwards by five units.
• Horizontally - through the adjustment in the x-term, though in this problem, the horizontal translation is not present.

Translations are straightforward yet powerful tools, as they allow us to modify the position of graphs on a coordinate plane while maintaining their original shape. It's crucial to remember that these shifts do not affect the function's symmetry or the direction the parabola opens.

Vertex Shifting

Vertex shifting refers to the movement of the vertex of a quadratic function's graph due to function translation. The vertex is the point where the parabola changes its direction. For the standard quadratic function \( g(x) = x^2 \), the vertex is at (0, 0).

• In \( f(x) = x^2 + 5 \), the additional "5" moves the vertex from (0, 0) to (0, 5), indicating a vertical shift.
• The vertex movement helps us easily identify how the graph has transformed without affecting its curvature or opening direction.

Understanding vertex shifting is crucial for sketching precise graphs of transformed functions. It provides a clear reference point and helps visualize how the entire graph is repositioned through simple transformations.