Problem 30 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of the equation. \(x=y^{2}+4\)

Step-by-Step Solution

Answer

The graph of the equation \(x=y^{2}+4\) is a parabola opening to the right, with a vertex at (-4, 0).

Step 1: Understand the Shape of the graph

Identify that the equation \(x=y^{2}+4\) represents a parabola that opens horizontally instead of vertically. This is because the 'y' is squared, not the 'x'.

Step 2: Find the Vertex

The vertex of this parabola is at the point (-4, 0). You can determine this by seeing that when y=0, x=4. However, since the equation is subtracted by 4 on the right, the parabola is shifted 4 units to the left, resulting in a vertex of (-4,0).

Step 3: Sketching the points

Pick a few points above and below y=0 to see the shape of the parabola. For example, when y=1 and y=-1, x=5. When y=2 and y=-2, x=8. So plot the points (5,1), (5,-1), (8,2), and (8,-2) along with the vertex (-4,0).

Step 4: Draw the Parabola

Now draw a smooth curve through these points. The graph opens horizontally to the right, confirming our initial identification.

Key Concepts

Horizontal Parabola

Understanding the orientation of a parabola is crucial for graphing. In the equation x = y^2 + 4, we identify the parabola's horizontal opening due to the squared term being the y, rather than the x. This differs from the more commonly seen vertical parabola where x is the squared term.

Horizontal parabolas such as this have their axis of symmetry parallel to the horizontal axis, or the x-axis, instead of the vertical y-axis. The directional opening of the parabola (left or right) is determined by the sign before the y^2 term; if it's positive, as it is here, the parabola opens to the right. If it were negative, the parabola would open to the left.

Vertex of a Parabola

The vertex of a parabola is the point where it turns; its most distinct feature. In the given equation x = y^2 + 4, the vertex can be found by setting y to zero since the square of zero is also zero. Here, when y = 0, we have x = 4. However, the presence of a '+' or '–' after the squared term indicates a horizontal shift. With x = y^2 + 4, the parabola is shifted 4 units left, placing the vertex at (-4, 0).

The vertex serves as a starting point when sketching a parabola and assists in understanding its width and direction.

Quadratic Equations

Quadratic equations are at the heart of parabolas. They take the standard form ax^2 + bx + c = 0 when expressing vertical parabolas, but can also describe horizontal ones as seen in x = y^2 + 4. The solution to these equations often results in two real values for x when graphed—it's where the parabola crosses the x-axis.

However, when couched in terms of another variable, as with y in this case, the equation still represents a quadratic relationship between x and y. This quadratic form underpins the characteristic ‘U’ shape of the parabola, whether it's oriented vertically or horizontally.

Sketching Graphs

To effectively sketch graphs of quadratic functions, begin with the vertex. For the equation x = y^2 + 4, we start at the vertex (-4, 0). Then, select values for y and substitute them into the equation to find corresponding x values; plotting these points provides the curvature shape.

Using symmetrical points about the vertex makes plotting easier. For instance, selecting y = 1 and y = -1 gives the same x (in this case 5), creating a mirrored effect. Once enough points are plotted, a smooth curve drawn through these points reveals the parabola. Remember that the graph of a quadratic equation is always a parabola, and, in sketching, consistency in curve smoothness and symmetry is key.