Problem 38 - Intermediate Algebra

Question

(a) Graph \(y=x^{2}, y=(x-2)^{2}+3, y=(x+4)^{2}-2\), and \(y=(x-6)^{2}-4\) on the same set of axes. (b) Graph \(y=x^{2}, y=2(x+1)^{2}+4, y=3(x-1)^{2}-3\), and \(y=\frac{1}{2}(x-5)^{2}+2\) on the same set of axes. (c) Graph \(y=x^{2}, y=-(x-4)^{2}-3, y=-2(x+3)^{2}-1\), and \(y=-\frac{1}{2}(x-2)^{2}+6\) on the same set of axes.

Step-by-Step Solution

Answer

Transform and graph each function based on transformations such as shifts, stretches, compressions, and reflections derived from \(y = x^2\).

Step 1a: Understand the Function Transformations

We start with the basic quadratic function, which is given by \(y = x^2\). Each transformed function is a variation of this basic form:- \(y = (x-2)^2 + 3\) is a horizontal shift 2 units right and a vertical shift 3 units up.- \(y = (x+4)^2 - 2\) is a horizontal shift 4 units left and a vertical shift 2 units down.- \(y = (x-6)^2 - 4\) is a horizontal shift 6 units right and a vertical shift 4 units down.

Step 2a: Graph Functions from Part (a)

To graph these functions:1. Start by plotting \(y = x^2\), which is the standard parabola opening upwards.2. For \(y = (x-2)^2 + 3\), shift each point of the standard parabola 2 units to the right and 3 units up.3. For \(y = (x+4)^2 - 2\), shift each point 4 units to the left and 2 units down.4. For \(y = (x-6)^2 - 4\), shift each point 6 units to the right and 4 units down. Plot all transformations on the same axis.

Step 1b: Understand the Function Transformations for Part (b)

These transformations are also based on \(y = x^2\):- \(y = 2(x+1)^2 + 4\) involves stretching vertically by a factor of 2, a horizontal shift left by 1, and up by 4.- \(y = 3(x-1)^2 - 3\) involves stretching by a factor of 3, shifting right by 1, and down by 3.- \(y = \frac{1}{2}(x-5)^2 + 2\) involves compressing by a factor of \(\frac{1}{2}\), shifting right by 5, and up by 2.

Step 2b: Graph Functions from Part (b)

Now plot for part (b):1. Graph \(y = x^2\) for reference.2. For \(y = 2(x+1)^2 + 4\), apply a vertical stretch, left shift by 1, and move up by 4.3. For \(y = 3(x-1)^2 - 3\), apply a vertical stretch, a right shift by 1, and move down by 3.4. For \(y = \frac{1}{2}(x-5)^2 + 2\), apply a vertical compression, right shift by 5, and move up by 2.

Step 1c: Understand the Function Transformations for Part (c)

Considerations for part (c) of the problem:- \(y = -(x-4)^2 - 3\) is a reflection over the x-axis, a right shift by 4, and down by 3.- \(y = -2(x+3)^2 - 1\) involves a reflection, stretching by 2, a left shift by 3, and down by 1.- \(y = -\frac{1}{2}(x-2)^2 + 6\) involves a reflection, compression by \(\frac{1}{2}\), a right shift by 2, and up by 6.

Step 2c: Graph Functions from Part (c)

For part (c), each function transforms the original \(y = x^2\) as follows:1. Draw the basic parabola \(y = x^2\) for orientation.2. Reflect and shift \(y = -(x-4)^2 - 3\) right and down.3. Reflect, stretch and shift \(y = -2(x+3)^2 - 1\) left and down.4. Reflect, compress and shift \(y = -\frac{1}{2}(x-2)^2 + 6\) right and up.

Key Concepts

Graphing Parabolas

A parabola is the graph of a quadratic function, typically in the form of \(y = ax^2 + bx + c\). The simplest form is \(y = x^2\), which produces a symmetrical curve opening upwards centered at the y-axis. To understand parabolas, it helps to familiarize yourself with the vertex, which is the highest or lowest point on the parabola and provides symmetry. The axis of symmetry is a vertical line that passes through the vertex.

When graphing, the standard parabola \(y = x^2\) serves as our "parent function." Each transformed quadratic can be derived from this base graph by applying specific transformations, such as shifts, reflections, stretches, and compressions. Transforming the graph helps us understand how parabolas shift position or change shape in the coordinate plane.

Horizontal and Vertical Shifts

Shifting a parabola involves moving the entire graph either horizontally or vertically without changing its shape. These shifts are achieved through adjustments to the function's equation.

• A horizontal shift occurs when the parabola moves left or right. In the equation \(y = (x-h)^2\), the term \(h\) indicates a shift. If \(h\) is positive, the shift is to the right, and if negative, it is to the left.
• A vertical shift moves the parabola up or down. This is controlled by \(k\) in the equation \(y = x^2 + k\). If \(k\) is positive, the graph moves upward; if negative, it moves downward.

For example, \(y = (x-2)^2 + 3\) is shifted 2 units to the right and 3 units up. These shifts do not alter the parabola's shape; they simply reposition it on the graph.

Function Reflections

Reflections involve flipping the graph of a parabola over a specific line, most commonly the x-axis. In quadratic functions, this is achieved by multiplying the function by -1.

• To reflect a parabola over the x-axis, you multiply the entire function by -1. For instance, \(y = -(x-4)^2\) flips the parabola of \(y = (x-4)^2\) upside down.

Reflections can change the parabola's direction (from opening upwards to downwards and vice versa), but the vertex and shape remain the same size, only mirrored. This transformation is essential when working with quadratics in maximizing or minimizing contexts, like physics problems involving trajectories.

Vertical Stretch and Compression

Vertical stretching and compression refer to altering the parabola's width or steepness. These transformations are essential for understanding the effect of the coefficient \(a\) in the quadratic function \(y = ax^2\).

• A vertical stretch happens when \(|a| > 1\). The parabola becomes more "narrow" or steep. For example, \(y = 3(x-1)^2 - 3\) is vertically stretched compared to \(y = x^2\).
• A vertical compression occurs when \(0 < |a| < 1\). This makes the parabola wider. For example, \(y = \frac{1}{2}(x-5)^2 + 2\) compresses the graph.

The sign of \(a\) determines if the parabola opens upwards (positive \(a\)) or downwards (negative \(a\)). Adjusting \(a\) while keeping the vertex form \(y = a(x-h)^2 + k\) helps visualize how much the parabola stretches or compresses. This manipulation is crucial for accurate graphing and understanding quadratic function behaviors in different scenarios.