Problem 40 - A Graphical Approach to Precalculus with Limits

Question

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. $$f(x)=\frac{1}{2}(x+2)^{2}$$

Step-by-Step Solution

Answer

The graph is a parabola opening upwards, shifted 2 units left, and vertically compressed by a factor of 2.

Step 1: Identify the Basic Graph

The given function is a transformation of the basic quadratic function \(y = x^2\). Identify that the basic graph shape of \(y = x^2\) is a parabola opening upwards with its vertex at the origin \((0,0)\).

Step 2: Apply Horizontal Shift

The term \((x+2)\) indicates a horizontal shift. Specifically, \(x\) is replaced with \(x+2\), which shifts the graph 2 units to the left. So, the new vertex of the parabola is now at \((-2,0)\).

Step 3: Apply Vertical Stretch and Compression

The factor \(\frac{1}{2}\) in front of \((x+2)^2\) is a vertical compression. This means the graph of the parabola becomes wider since any \(y\)-value for a given \(x\) is now half of what it would be in \(y=(x+2)^2\).

Step 4: Sketch the Transformed Graph

Start by marking the new vertex at \((-2,0)\). Since the graph opens upwards and is vertically compressed, sketch the parabola with points that reflect these transformations, ensuring that it is symmetric around the vertical line \(x=-2\).

Key Concepts

Quadratic Functions

Quadratic functions are fundamental mathematical expressions that take the form \( y = ax^2 + bx + c \). These functions represent a parabola when graphed on the coordinate plane. Their most basic form, \( y = x^2 \), creates a symmetrical curve that opens upwards. The shape and orientation of this curve can change depending on the values of the coefficients \( a \), \( b \), and \( c \). In the formula, \( a \) determines the direction and width of the parabola. If \( a \) is positive, the parabola opens upwards; if negative, it opens downwards. A larger value of \( a \) results in a narrower parabola, whereas a smaller \( a \) leads to a wider curve. This makes quadratic functions incredibly versatile in graph transformations.

Parabolas

Parabolas are U-shaped curves that represent the graph of a quadratic function. They have a distinct axis of symmetry, a vertical line that divides the parabola into two mirror-like halves. The vertex, located on the axis of symmetry, is a critical point where the parabola changes direction. The standard parabola \( y = x^2 \) has its vertex at \( (0, 0) \). This point is also the minimum point if the parabola opens upwards and the maximum if it opens downwards.

• The direction in which the parabola opens depends on the sign of the leading coefficient \( a \).
• The vertex is a vital reference in graph transformations, serving as a starting point for shifts and stretches.

Understanding these characteristics of parabolas is crucial to predicting how they transform under various operations.

Horizontal Shift

A horizontal shift in a graph involves moving the entire graph left or right on the coordinate plane without altering its shape. For a quadratic function \( y = (x-h)^2 \), the term \( h \) dictates the horizontal shift. If \( x \) is replaced with \( x + 2 \), as in the exercise \( y = \frac{1}{2}(x+2)^2 \), the graph shifts 2 units to the left. This change in position redefines the location of the vertex from \( (0, 0) \) to \( (-2, 0) \). During a horizontal shift,

• The symmetry and orientation of the parabola remain unchanged.
• The axis of symmetry moves along with the vertex, helping to maintain the parabola's symmetry.

Horizontal shifts are straightforward yet impactful in altering a graph's appearance.

Vertical Compression

Vertical compression involves altering the vertical scale of a graph, effectively compressing the shape without changing its position on the horizontal axis. This type of transformation is indicated by a coefficient multiplying the function. In the expression \( y = \frac{1}{2}(x+2)^2 \), the \( \frac{1}{2} \) compresses the parabola vertically, making the graph appear wider compared to the standard \( y = x^2 \).

• A vertical stretch would occur if the multiplying factor were greater than 1, making the parabola narrower.
• The vertex remains in place during vertical compression, ensuring the graph's minimum or maximum point stays constant.

Understanding the effects of vertical compression is essential for anticipating how changes in the multiplicative coefficient affect the shape of the graph.