Problem 14 - The Calculus with Analytic Geometry

Question

In Exercises 7 through 28 , draw a sketch of the graph of the equation. $$ x=y^{2}+1 $$

Step-by-Step Solution

Answer

Plot the vertex at (1, 0) and several points using the equation, then draw a curve through them to form a parabola opening to the right.

Step 1: Understand the Equation

The given equation is \[ x = y^2 + 1 \]. This represents a parabola. Note that it is a parabola that opens along the x-axis because the variable y is squared.

Step 2: Identify the Vertex

The vertex of the equation \[ x = y^2 + 1 \] can be found by setting \[ y = 0 \]. This gives \[ x = 0^2 + 1 = 1 \]. Therefore, the vertex is at the point (1, 0).

Step 3: Determine the Direction

Since the coefficient of \[ y^2 \] is positive, the parabola opens to the right along the x-axis.

Step 4: Plot Key Points

Select some values of \[ y \] to find corresponding \[ x \] values. For example: \[ y = 1 \] gives \[ x = 1^2 + 1 = 2 \] resulting in the point (2, 1). \[ y = -1 \] gives \[ x = (-1)^2 + 1 = 2 \] resulting in the point (2, -1). Do this for a few more points: \[ y = 2 \] gives \[ x = 2^2 + 1 = 5 \] corresponding to (5, 2), and \[ y = -2 \] gives \[ x = 5 \] corresponding to (5, -2).

Step 5: Sketch the Graph

Plot the vertex (1, 0) and the points (2, 1), (2, -1), (5, 2), and (5, -2) on a coordinate plane. Draw a smooth curve through these points to form the parabola that opens to the right.

Key Concepts

parabolas

A parabola is a symmetrical, curved shape that forms when graphing quadratic equations. A basic quadratic equation is typically represented as either \[ y = ax^2 + bx + c \] or \[ x = ay^2 + by + c \], where 'a,' 'b,' and 'c' are constants. The equation determines the direction and openness of the curve. If 'a' is positive, the parabola opens upward (or to the right for functions involving y); if 'a' is negative, it opens downward (or to the left).

In the given exercise, the parabola is represented by the equation \[ x = y^2 + 1 \]. Here, 'a' is 1, so the parabola opens to the right since the y term is squared. Understanding these properties helps in graphing parabolas accurately.

vertex of a parabola

The vertex is a key point on the parabola that represents its highest or lowest point. For parabolas that open along the x-axis, like in the equation \[ x = y^2 + 1 \], the vertex can be found by setting the non-squared variable (in this case, y) to zero.

Setting \[ y = 0 \] in our equation gives: \[ x = 0^2 + 1 = 1 \].

Therefore, the vertex is at (1, 0). This point acts as a reference to help plot the rest of the graph. Knowing the vertex allows us to sketch the axis of symmetry—the imaginary line that divides the parabola into two mirror-image halves.

plotting points on a graph

Once you've identified the vertex, the next step is to find additional points to help outline the curve of the parabola. This is done by selecting values for 'y' and calculating the corresponding 'x' values. Here’s how:

- For \[ y = 1 \]: \[ x = 1^2 + 1 = 2 \] results in the point (2, 1).
- For \[ y = -1 \]: \[ x = (-1)^2 + 1 = 2 \] results in the point (2, -1).
- For \[ y = 2 \]: \[ x = 2^2 + 1 = 5 \] results in the point (5, 2).
- For \[ y = -2 \]: \[ x = 5 \] results in the point (5, -2).

Plot these points on a graph to form a clear shape and ensure the parabola is correctly drawn.

coordinate geometry

Coordinate geometry, or analytic geometry, is the study of geometry using a coordinate system. This system helps in solving geometric problems by placing shapes in a more calculated and graphical form. Key elements include:

- **Coordinate plane**: A two-dimensional plane formed by the x-axis (horizontal) and y-axis (vertical), meeting at the origin (0,0).
- **Points**: Represented as (x, y), points are locations on the coordinate plane.

In the context of this exercise, each calculated point (like (2, 1) or (5, -2)) is plotted on the coordinate plane to sketch the parabola. Through coordinate geometry, we can visualize and solve equations systematically, making abstract algebraic concepts easier to understand.