Problem 49 - College Algebra and Calculus: An Applied Approach

Question

Use symmetry to complete the graph of the equation. \(y\) -axis symmetry \(y=-x^{2}+4\)

Step-by-Step Solution

Answer

The graph of \(y=-x^{2}+4\) is a parabola opening downwards, with vertex at (0,4) and symmetric about the y-axis. Its other two points are (1,3) and (-1,3), obtained by evaluating the function for the values \(x=1\) and \(x=-1\) respectively.

Step 1: Identify the Symmetry

We are told that the graph of the equation has y-axis symmetry. This implies that for every point \(P(x,y)\) on the graph, the point \(P'(-x,y)\) is also on the graph.

Step 2: Plot the Vertex and Axis of Symmetry

The vertex of the quadratic \(y=-x^{2}+4\) is the maximum point as the coefficient of \(x^{2}\) is negative, and occurs at \(x=0\). For \(x=0\), \(y= - (0)^{2}+4 =4\), so the vertex is at the point (0,4). Draw a dotted vertical line, the y-axis in this case, through the vertex to represent the axis of symmetry.

Step 3: Plot Other Points and Reflect

Plot points on either side of the axis of symmetry. For example, for \(x=1, y = - (1)^{2}+4 =3\) and for \(x=-1, y = -(-1)^{2}+4 =3\). As the graph has y-axis symmetry, we reflect each point on the right of the y-axis over to the left.

Step 4: Sketch the Complete Graph

Join these points to form a smooth curve, completing the parabola. Verify that it is a downward-opening parabola due to the negative coefficient of \(x^{2}\), and symmetric about the y-axis.

Key Concepts

Graph Symmetry

In mathematics, symmetry plays a vital role in simplifying graph construction and analysis. When a graph exhibits symmetry, it reflects a balanced and proportional distribution of points across its axis. For a quadratic function like \( y = -x^2 + 4 \), symmetry can be identified with respect to the y-axis.

Characteristics of y-Axis Symmetry:

• A graph having y-axis symmetry means for every point \((x, y)\) on the graph, there exists a corresponding point \((-x, y)\).
• Each point on one side of the y-axis has a mirror image on the opposite side.
• This reflection helps in plotting less number of points to realize the complete graph of a function.

By identifying this property, we can easily sketch quadratic functions. In the given example, recognizing the y-axis symmetry allowed us to mirror points and flesh out the complete parabola efficiently.

Vertex of a Parabola

The vertex is a crucial concept when graphing a parabola. Specifically, in the equation \( y = -x^2 + 4 \), the vertex is the highest or lowest point, depending on the parabola's orientation. Here, it is a maximum point because the coefficient of \( x^2 \) is negative.

For this parabola, the vertex can be found by analyzing the equation:

• The formula of a quadratic function is typically \( y = ax^2 + bx + c \). The vertex's x-coordinate is determined by \( x = -\frac{b}{2a} \).
• In our case, since \( b = 0 \), the vertex is at \( x = 0 \).
• Plugging this value into the equation gives us \( y = -(0)^2 + 4 = 4 \), making the vertex \( (0, 4) \).

Having identified the vertex, we know the point where the graph changes direction, aiding in constructing the parabola correctly.

Axis of Symmetry

The axis of symmetry is an invisible line that runs vertically through the center of a parabola, dividing it into two identical halves. For any quadratic function, particularly \( y = -x^2 + 4 \), this concept is vital for understanding how the graph is structured.

Determining the Axis of Symmetry:

• The axis of symmetry in any parabola of the form \( y = ax^2 + bx + c \) can be found at \( x = -\frac{b}{2a} \).
• In our example, since \( b = 0 \) and \( a \) is negative, the axis of symmetry is simply the y-axis or \( x = 0 \).
• This axis allows us to reflect points from one side to the other, confirming that the graph is complete and well-proportioned.

Understanding this guides not only in plotting but also in comprehending the overall symmetry and balance of quadratic functions.