Problem 65 - 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. $$ h(x)--2(x+1)^{2}+1 $$

Step-by-Step Solution

Answer

The graph of \(h(x) = 2(x+1)^2 +1\) is a parabola that opens upwards, with a vertex at (-1, 1) due to a shift one unit left and one upwards, and is stretched vertically by a factor of 2.

Step 1: Graph the standard quadratic function

Begin by graphing \(f(x) = x^2\). This is a simple parabola that opens upwards and has its vertex at the origin (0,0).

Step 2: Identify the transformations based on the new function

The given function is \(h(x) = 2(x+1)^2 +1\). This function can be described as a transformation of the standard function in the following ways: a shift to the left by one unit (represented by the \(x+1) term, a vertical stretch by factor of 2 (represented by the factor of 2 in front of the quadratic term), and a shift up by one unit (represented by the +1 at the end of the expression).

Step 3: Apply the transformations to the standard function

Start with the basic graph of \(f(x)=x^2\). Apply the identified transformations in order. Begin by shifting the graph one unit to the left - this corresponds to replacing every x in the original graph with \(x+1\). Next, stretch the graph vertically by a factor of 2. Finally, shift the graph upwards by one unit to represent the +1.

Key Concepts

Graph Transformations

Graph transformations help us modify a function's graph systematically. They make it easier to understand how our function changes with new parameters. In our problem, starting from the basic quadratic function \[ f(x) = x^2 \] we apply transformations one by one.

• Shifting: The expression \((x+1)\) inside the quadratic indicates a horizontal shift. Specifically, it shifts the graph to the left by one unit. Essentially, you're moving the entire parabola a little to the left.
• Vertical Stretch: The factor of 2 in front of \((x+1)^2\) stretches the graph vertically. Each point on the original parabola doubles its distance from the x-axis. Thus, making the parabola "narrower" as it stretches up or down.
• Vertical Shift: Adding 1 outside the squared term shifts the whole graph up by one unit. It means every point on the parabola moves one unit higher, adjusting the vertex and all other points up.

By applying these transformations in sequence, you get the graph of the new function \( h(x) = 2(x+1)^2 + 1 \).

Parabolas

Parabolas are the graphical representation of quadratic functions. They are U-shaped and can open upwards or downwards. The standard parabola you might be familiar with is from the quadratic function \[ f(x) = x^2 \]This basic form opens upwards, and its vertex, the point where it turns, is at the origin (0,0). But not all parabolas are created equal! Various transformations can shift, stretch, or compress them.

• Vertex: This is the highest or lowest point on a parabola, depending on its direction. For \( f(x) = x^2 \), the vertex is (0,0). After transformations, the vertex of the function \( h(x) = 2(x+1)^2 + 1 \) moves to (-1,1).
• Direction: The direction in which a parabola opens is determined by the sign of the coefficient in front of the squared term. A positive sign means it opens upwards; a negative sign flips it downwards. For our transformed function, the coefficient 2 (positive) indicates it opens upwards.

Understanding these properties helps in graphing more complex parabolas and anticipating how they will look after applying transformations.

Vertex Form

Vertex form is a convenient way to express a quadratic function, making it easier to identify the parabola's vertex. It generally looks like this:\[ y = a(x-h)^2 + k \]Where (h, k) is the vertex of the parabola.

• Components:
  • \(a\): This value determines how "wide" or "narrow" the parabola is, and whether it opens upwards or downwards. Larger absolute values make the parabola narrower.
  • \(h\): Shifts the parabola left or right. It moves the vertex horizontally to \(x = h\).
  • \(k\): Moves the parabola up or down, shifting the vertex to \(y = k\).
• Vertex Position: From our exercise, \( h(x) = 2(x+1)^2 + 1 \) is in vertex form, where \( a = 2 \), \( h = -1 \), and \( k = 1 \). The vertex therefore, is at (-1,1).

By recognizing a quadratic in vertex form, you can quickly determine key characteristics and graph the parabola efficiently. It's a great shortcut for visualizing transformations and understanding how the quadratic behaves.