Problem 89 - Calculus Volume 1

Question

Use the graph of \(f(x)=x^{2}\) to graph each transformed function \(g\). \(g(x)=(x+3)^{2}+1\)

Step-by-Step Solution

Answer

The graph of \(g(x) = (x+3)^2 + 1\) is a parabola with its vertex at (-3,1), shifted left by 3 units and up by 1 unit from the parent function.

Step 1: Identify the Parent Function

The parent function for the given transformation is \(f(x) = x^2\). This is a standard parabola that opens upwards with its vertex at the origin, (0,0).

Step 2: Determine Horizontal Shift

The function \(g(x) = (x+3)^2+1\) includes the term \((x+3)^2\). This indicates a horizontal shift to the left by 3 units from the origin. This shifts the vertex from (0,0) to (-3,0).

Step 3: Determine Vertical Shift

The function has the additional constant term \(+1\), which indicates a vertical shift upwards by 1 unit. This shifts the vertex from (-3,0) to (-3,1).

Step 4: Plot the Vertex

From the transformations determined, plot the new vertex at (-3,1) on the graph.

Step 5: Sketch the Parabola

Since there are no stretches or reflections, the parabola retains the same shape as the parent function \(f(x) = x^2\). Draw a smooth curve opening upwards with its new vertex at (-3,1). Ensure it is symmetrical around the vertical line \(x = -3\).

Key Concepts

Horizontal Shift

When functions undergo a horizontal shift, the entire graph of the function moves horizontally along the x-axis. In our exercise, the function \(g(x)=(x+3)^2+1\) demonstrates a horizontal shift. The term \((x+3)^2\) indicates a shift to the left by 3 units.

To understand why it shifts to the left, consider that if there was no horizontal shift, the term would be in the form \((x-0)^2\). Changing it to \(x+3\) means adjusting for the opposite, moving the vertex from \((0,0)\) to \((-3,0)\). This principle holds generally:

• If you see \((x-a)^2\), the graph shifts to the right by \(a\) units.
• If you see \((x+a)^2\), the graph shifts to the left by \(a\) units.

Grasping horizontal shifts is fundamental as it helps predict new vertex positions in a transformed function.

Vertical Shift

Vertical shifts in function graphs occur when you add or subtract a constant value, resulting in the graph moving up or down on the y-axis. For \(g(x)=(x+3)^2+1\), the term \(+1\) indicates a vertical shift.

In this scenario, the entire graph moves up by 1 unit. This means that our vertex, previously at \((-3,0)\) after the horizontal shift, is further elevated to \((-3,1)\). Vertical shifts are straightforward:

• A positive constant adds height to each point on the graph, moving it upwards.
• A negative constant lowers each point, pulling the graph downwards.

Understanding vertical shifts is crucial for mapping graphs accurately and predicting function behavior with transformations.

Parabola

A parabola is a U-shaped curve that is the graph of a quadratic function like \(f(x)=x^2\). It's defined by its symmetry and the location of its vertex. In its standard form, the parabola \(y=x^2\) has its vertex at the origin and opens upwards.

In function transformations, like in \(g(x)=(x+3)^2+1\), the basic shape of the parabola doesn't change unless there's a reflection or stretch (which isn't present here). The key modifications involve moving or rotating the parabola, but its curvature remains consistent.

• For upward-opening parabolas, like \(y = x^2\), the endpoints go to infinity as x moves away from the vertex.
• The parabola is always symmetric around a vertical line that passes through its vertex.

Understanding parabolas is crucial as they frequently appear in math and science, modeling everything from projectile motion to economic graphs.

Vertex

The vertex of a parabola is an essential reference point, marking the curve's extreme point. For the parent function \(f(x)=x^2\), the vertex is at \((0,0)\). For transformed functions, such as \(g(x)=(x+3)^2+1\), identifying the vertex is part of understanding the overall shift and position of the parabola.

The transformations in the exercise shifted the vertex first to \((-3,0)\) through a horizontal shift, then to \((-3,1)\) with a vertical shift. This allows us to clearly define the new position of the parabola's peak or trough, which is crucial for accurate graphing.

• The vertex provides symmetry; for any point \(a\) units to the left of the vertex, there's a corresponding point \(a\) units to the right.
• In real-world applications, the vertex can represent maximum or minimum values, like the highest point of a thrown ball.

Recognizing and accurately placing the vertex helps in both sketching and interpreting the parabolic graph effectively.