Problem 39 - 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. $$ y=3-\frac{1}{2}(x-1)^{2} $$

Step-by-Step Solution

Answer

It's a downward-opening parabola, vertex at (1, 3).

Step 1: Identify the Base Function

The given function is \[ y = 3 - \frac{1}{2}(x-1)^2 \]The base function here is the standard quadratic function \[ y = x^2 \] This base function is a parabola that opens upwards, with its vertex at the origin (0, 0).

Step 2: Apply Horizontal Shift

The expression \((x-1)^2\) inside the function indicates a horizontal shift. This means the parabola is shifted right by 1 unit. So, the new vertex is at (1, 0).

Step 3: Apply Vertical Stretch/Compression

The coefficient \(-\frac{1}{2}\) in front of the \((x-1)^2\) term indicates a vertical compression and reflection. The reflection is because the coefficient is negative, meaning the parabola will open downwards. The compression reduces the rate at which the parabola opens from its vertex.

Step 4: Apply Vertical Shift

The \(+3\) at the beginning of the function, \(y = 3 - \frac{1}{2}(x-1)^2\), shifts the whole parabola up by 3 units. So the final vertex of the parabola is located at (1, 3).

Step 5: Sketch the Final Graph

Combine all transformations: Start with the standard parabola \(y = x^2\), shift it right by 1 unit, reflect and compress it vertically, and then shift it up by 3 units. The final graph is a downward-opening parabola with its vertex at (1, 3).

Key Concepts

Quadratic Function

Understanding a quadratic function is key to mastering many topics in algebra. A quadratic function is typically written in the form \[ y = ax^2 + bx + c \]where \(a\), \(b\), and \(c\) are constants. The most basic form, \( y = x^2 \), is a parabola, a U-shaped curve that opens upwards from the vertex at the origin (0,0).

• The coefficient \( a \) determines the direction and width of the parabola. If \( a \) is positive, it opens upwards; if negative, it opens downwards.
• In the vertex form \( y = a(x-h)^2 + k \), the vertex of the parabola is at \((h, k)\). This form makes it easier to identify shifts and transformations.

Recognizing this base function makes it simpler to apply any transformations like vertical or horizontal shifts.

Vertical Shift

A vertical shift in a function involves moving the graph up or down without changing its shape. For a quadratic function like \( y = (x-h)^2 + k \), the \(k\) value shifts the parabola.

• If \(k\) is positive, the parabola shifts up.
• If \(k\) is negative, the parabola shifts down.

In the given function \( y = 3 - \frac{1}{2}(x-1)^2 \), the \(+3\) indicates a vertical shift upward. This means every point on the parabola goes 3 units higher. Therefore, if the vertex of the standard parabola \( y = x^2 \) is at (1,0), shifting it vertically makes the new vertex (1,3). This transformation helps visualize where the graph starts on the y-axis.

Horizontal Shift

Horizontal shifts move the graph left or right along the x-axis. The form \( y = (x-h)^2 \) directly indicates horizontal shifts. The value of \(h\) tells you how far and in what direction to move the parabola:

• If \(h\) is positive, the shift is to the right.
• If \(h\) is negative, the shift is to the left.

In our exercise, \( (x-1) \) signifies a shift of 1 unit to the right. This alters the vertex of \( y = x^2 \) from the origin (0,0) to (1,0). It’s crucial to adjust the vertex position first to accurately apply other transformations like vertical shifts or reflections.

Parabola Reflection

Reflection in a parabola occurs when it flips over an axis. For quadratic functions, a negative coefficient in front of the squared term indicates that the parabola opens in the opposite direction. For example:

• A positive \(a\) makes the parabola open upwards \( y = ax^2 \).
• A negative \(a\) makes it open downwards, reflecting over the x-axis.

In the equation \( y = 3 - \frac{1}{2}(x-1)^2 \), the term \(-\frac{1}{2}\) tells us the parabola is reflected down. Additionally, it leads to a vertical compression, making the parabola less steep than the standard one. Understanding the effect of this negative sign helps in visualizing the direction in which the parabola will open and how it will appear on a graph.