Problem 7 - Algebra A Combined Function

Question

Graph each inequality. $$ y>(x-1)^{2}-3 $$

Step-by-Step Solution

Answer

Graph a dashed parabola \( y = (x-1)^2 - 3 \) and shade the region above it.

Step 1: Understand the Inequality

The inequality is given as \( y > (x-1)^2 - 3 \). This represents a region above the parabola given by the equation \( y = (x-1)^2 - 3 \), where the parabola opens upwards.

Step 2: Identify the Shape of the Graph

The given expression \( (x-1)^2 - 3 \) represents a parabola. Since the expression involves \( (x-1)^2 \), the parabola is centered at \( x = 1 \) and is shifted downward by 3 units due to the \(-3\).

Step 3: Sketch the Parabola

Start by sketching the parabola described by the equation \( y = (x-1)^2 - 3 \). This is a basic parabola with vertex at \((1, -3)\) and line of symmetry parallel to the y-axis. The parabola opens upwards.

Step 4: Determine the Exclusion of the Boundary

Since the inequality is strict (\( y > (x-1)^2 - 3 \)), the boundary line represented by \( y = (x-1)^2 - 3 \) is not included in the solution. This means the parabola will be drawn as a dashed line to indicate this exclusion.

Step 5: Shade the Appropriate Region

Identify where \( y \) is greater than \((x-1)^2 - 3\) by shading the region above the dashed parabola. This region represents all the points where the inequality is satisfied.

Key Concepts

Parabolas

Parabolas are one of the most common quadratic graph shapes you will encounter in mathematics. They have a distinct 'U' shape and can open upwards or downwards.
Understanding the features of a parabola can help you easily visualize its graph.

• The "vertex" of the parabola is its turning point. When the parabola opens upwards, the vertex is the lowest point.
• The "line of symmetry" is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves.
• Parabolas can be shifted up, down, left, or right by adding or subtracting numbers inside the function.

In the given inequality, the parabola is centered at \(x = 1\) and opens upwards. It is a standard form parabola that has just been shifted down by 3 units.

Vertex Form

The vertex form of a parabola provides valuable information about its graph, especially location and shape.
A parabola in vertex form generally looks like this: \[ y = a(x-h)^2 + k \]where:

• \((h, k)\) is the vertex of the parabola.
• \(a\) determines how "wide" or "narrow" the parabola is. If \(a\) is positive, the parabola opens upward, and if it's negative, it opens downward.

Using the vertex form, you can quickly locate the vertex. For example, in our equation \( y = (x-1)^2 - 3 \), the vertex is at \((1, -3)\). This makes sketching the graph much simpler because it gives us the starting point for drawing.

Graph Shading

Graph shading is essential for depicting solutions to inequalities. When you need to show that one side of a curve fulfills the inequality, you shade the region where the inequality is true.
In our example, because the inequality is \( y > (x-1)^2 - 3 \), this means any point \( (x, y) \) that lies above the parabola's curve satisfies the inequality.

• To show this, the region above the parabola is shaded.
• Shaded regions represent all the points where the inequality holds true.

Always remember, the shaded area indicates that any point within this space is a viable solution.

Inequality Symbols

Inequality symbols play a crucial role in determining the portions of the graph to include or exclude. They help specify which values of \(y\) are part of the solution set.
These symbols can be:

• \(>\) Greater than
• \(<\) Less than
• \(\geq\) Greater than or equal to
• \(\leq\) Less than or equal to

In our problem, \(y > (x-1)^2 - 3\), the "greater than" sign indicates that \(y\) must be more than the value of the parabola, excluding the actual boundary. So, we use a dashed line to show this boundary exclusion in the graph.