Problem 51 - College Algebra with Modeling and Visualization

Question

Use transformations to sketch a graph of \(f\). \(f(x)=-x^{2}+4\)

Step-by-Step Solution

Answer

Reflect the base parabola downwards and shift it up by 4 units.

Step 1: Identify the Base Function

The base function is the quadratic function \(g(x) = x^2\). This is a standard parabola opening upwards and centered at the origin.

Step 2: Reflect Across the X-axis

The function \(f(x) = -x^2 \) involves a reflection of the base function across the x-axis. This transforms the parabola to open downwards while maintaining the vertex at the origin.

Step 3: Vertical Shift Upwards

The term \(+4\) in the function \(f(x) = -x^2 + 4\) indicates a vertical shift upwards by 4 units. This moves the vertex of the parabola from the origin to the point (0, 4).

Step 4: Sketch the Graph

Begin with a downward-opening parabola whose vertex is at (0, 4). Plot a few points, such as (1, 3) and (-1, 3), to help guide the shape of the parabola and ensure symmetry about the vertex.

Key Concepts

Quadratic Function

A quadratic function is an essential concept in algebra that is commonly expressed in the form \(g(x) = ax^2 + bx + c\). The graph of a quadratic function is a parabola, which is a U-shaped curve. Three key features of a quadratic function are:

• The coefficient \(a\): If \(a > 0\), the parabola opens upwards; if \(a < 0\), it opens downwards.
• The axis of symmetry: This is a vertical line passing through the vertex of the parabola, given by \(x = -\frac{b}{2a}\).
• The vertex: This is the highest or lowest point on the graph of the parabola, located at \((-\frac{b}{2a}, f(-\frac{b}{2a}))\).

For the specific exercise described, the base quadratic function is \(g(x) = x^2\), meaning it is centered at the origin (0,0) and opens upwards. Understanding these features allows for easy identification and manipulation of quadratic functions.

Reflection Transformation

Reflection transformation is a type of graph transformation in which a function is flipped over a line, such as the x-axis or y-axis, producing a mirror image of the original graph.

• Reflection across the X-axis: This transforms the graph of \(y = f(x)\) into \(y = -f(x)\). The reflection reverses the direction in which the parabola opens, converting upward-opening functions to downward-opening, and vice-versa.
• Reflection across the Y-axis: For even functions \(y = f(x)\), this transformation doesn't change the graph since it remains symmetric about the y-axis.

In our exercise, reflecting the base function \(g(x) = x^2\) across the x-axis produces \(f(x) = -x^2\). This reflection turns the parabola to open downwards while keeping the vertex at the same position, the origin. Through reflections, we can effectively explore new graph orientations without altering the fundamental shape of the parabola.

Vertical Shift

A vertical shift is a simple graph transformation that involves moving the entire graph of a function up or down in the Cartesian plane without changing its shape.

• Upward Shift: Adding a constant \(k\) to a function \(y = f(x)\) results in \(y = f(x) + k\), shifting the graph upwards by \(k\) units.
• Downward Shift: Subtracting a constant \(k\) from a function \(y = f(x)\) creates \(y = f(x) - k\), shifting the graph downwards by \(k\) units.

Applying this to our exercise, after reflecting the quadratic function to get \(f(x) = -x^2\), adding 4 to the equation, as in \(f(x) = -x^2 + 4\), results in a vertical shift 4 units upwards. This moves the vertex of the parabola from the origin (0, 0) to the new point (0, 4). Vertical shifts facilitate positioning graphs appropriately without affecting the graph's structure.