Problem 53 - College Algebra

Question

Begin by graphing the standard quadratic function, \(f(x)-x^{2} .\) Then use transformations of this graph to graph the given function. $$ g(x)=x^{2}-2 $$

Step-by-Step Solution

Answer

The graph of the function \(g(x) = x^{2} - 2\) will be a parabola opening upwards and shifted 2 units down from the vertex of the standard function \(f(x) = x^{2}\). The vertex of the graph of \(g(x)\) is at point (0, -2).

Step 1: Graph the standard quadratic function

Begin by plotting the standard quadratic function \(f(x) = x^{2}\). This will be a parabola with a vertex at the origin (0,0), opening upwards, and symmetric about the y-axis.

Step 2: Identify the transformation

On observing the given function \(g(x) = x^{2} - 2\), we can see that it is the same as the standard function, but there is a '-2'. This is a vertical shift or translation, 2 units downwards.

Step 3: Apply the transformation and graph \(g(x)\)

Graph \(g(x) = x^{2} - 2\) by taking the original graph \(f(x) = x^{2}\) and shifting every point 2 units downwards. The vertex will now be at point (0, -2)

Key Concepts

Graph Transformations

Graph transformations involve changing the position or shape of a graph. These transformations include shifts, reflections, stretches, and compressions. For quadratic functions like parabolas, understanding transformations helps in sketching graphs effectively. Starting with the standard quadratic function, such as \(f(x) = x^{2}\), allows us to predict how its graph will transform with alterations. A positive or negative sign, even multiplication, affects the graph's orientation or width. Simple addition or subtraction applies shifts vertically or horizontally.

• Vertical shifts: Modify the y-values, moving the graph up or down.
• Horizontal shifts: Adjust the x-values, displacing the graph left or right.
• Reflections: Flip the graph over the x- or y-axis.
• Stretches/Compressions: Change how wide or narrow the graph appears.

Mastering these transformations is essential for analyzing and interpreting the graphical representation of functions.

Parabolas

Parabolas are symmetrical, U-shaped graphs of quadratic functions, such as \(f(x) = x^{2}\). They typically appear in the context of standard form equations, \(y = ax^{2} + bx + c\). The shape and direction of a parabola depend on the coefficient \(a\). For instance:

• If \(a > 0\), the parabola opens upwards, similar to \(f(x) = x^{2}\).
• If \(a < 0\), it opens downwards.

Parabolas have a vertex that serves as the highest or lowest point, depending on their orientation. The vertex's location is critical as it helps in understanding the parabolic shape and determining any transformations applied. Parabolas are always symmetric around a vertical line, known as the axis of symmetry. Understanding parabolas is crucial when graphing quadratic functions, as they represent the visual foundation of these equations.

Vertical Translation

Vertical translation is a key transformation affecting the placement of a function's graph along the y-axis. In simple terms, a vertical translation will move the entire graph up or down without altering its shape. Given the function \(g(x) = x^{2} - 2\), the '-2' indicates a vertical translation moving the parabola 2 units downward from its original position.

• The vertex of the function initially at \((0,0)\) shifts to \((0,-2)\).
• Every point on the graph moves equally downwards.
• The overall shape of the parabola remains unaltered.

It's essential to recognize vertical translation as it allows for quick adjustments to the graph's height, facilitating the graphical analysis of quadratic equations. Grasping this concept also aids in interpreting real-world scenarios modeled by quadratic functions.