Problem 61 - A Graphical Approach to Precalculus with Limits

Question

Each equation in Exercises defines a parabola. Without actually graphing. match each equation with the appropriate description. $$(y-4)^{2}=x+2$$ A. Vertex \((2,-4) ;\) opens downward B. Vertex \((2,-4) ;\) opens upward C. Vertex \((4,-2)\); opens downward D. Vertex \((4,-2)\); opens upward E. Vertex \((-2,4)\); opens left F. Vertex \((-2,4)\); opens right G. Vertex \((-4,2) ;\) opens left H. Vertex \((-4,2)\); opens right

Step-by-Step Solution

Answer

The equation \((y-4)^2 = x + 2\) corresponds to option F: Vertex (-2,4); opens right.

Step 1: Identify the Standard Form of the Equation

The given equation is \[(y-4)^2 = x + 2\].This equation resembles the standard form of a parabola that opens either to the left or to the right, which is \[(y-k)^2 = 4p(x-h)\], where \(h, k\) is the vertex.

Step 2: Determine the Vertex

By comparing the equation \[(y-4)^2 = x + 2\] with the standard form \[(y-k)^2 = 4p(x-h)\],we identify \(k = 4\) and \(h = -2\). Therefore, the vertex is at \((-2, 4)\).

Step 3: Analyze the Direction of Opening of the Parabola

Since the equation is in the form \((y-k)^2 = 4p(x-h)\), and the coefficient of \(x\) is positive (since it is \(x + 2\), which implies a positive coefficient for \(x\)), this parabola opens to the right.

Key Concepts

Vertex

The vertex of a parabola is an essential concept in understanding its shape and position. It's like the starting point or the peak of the curve. For the equation \((y-4)^2 = x+2\), which is written in a special way, we have to compare it to the standard form \((y-k)^2 = 4p(x-h)\). This helps us find the vertex. In this case, the equation shows that \(k = 4\) and \(h = -2\). Hence, the vertex is at \((-2, 4)\). This means the starting or the turning point of the parabola is at \(x = -2\) and \(y = 4\). Understanding the vertex helps us imagine where the parabola is located on the graph.

Direction of opening

The direction a parabola opens tells us how it spreads out. It can open up, down, left, or right. For a parabola that opens sideways, we use the form \((y-k)^2 = 4p(x-h)\). In this form, the direction is influenced by the value of \(p\). If \(p\) is positive, the parabola opens to the right. If \(p\) is negative, it opens to the left.

• In our equation \((y-4)^2 = x + 2\), the parabola opens to the right.
• The positive association with \(x\), namely \(x + 2\), indicates a positive \(p\).

Understanding the direction of opening helps visualize how the parabola stretches across the graph.

Standard form of a parabola

The standard form of a parabola is crucial for identifying its key features without graphing. There are two main types of parabolas based on their directional opening: vertical and horizontal.

• Vertical parabolas follow the form \(y = ax^2 + bx + c\).
• Horizontal parabolas, like in our exercise, use \((y-k)^2 = 4p(x-h)\).

The equation \((y-4)^2 = x + 2\) is a horizontal parabola, which means it opens sideways.
This form directly helps identify the vertex \((h, k)\) and tells you the direction of the parabola, based on the sign of \(p\). Understanding this form makes it easier to find key points and describe the parabola without needing a graph.