Problem 11 - Algebra 2

Question

Graph each function. Identify the axis of symmetry. $$ y=-(x-7)^{2}+10 $$

Step-by-Step Solution

Answer

The graph is a downward-opening parabola with the vertex at (7, 10), and the axis of symmetry is the vertical line x = 7.

Step 1: Graphing the Function

To graph the function, first plot the vertex of the parabola. Since the function is in vertex form, \(y = a(x-h)^2 + k\), the vertex is at the point \(h,k\). For this function, \(h = 7\) and \(k = 10\), so the vertex is \( (7, 10) \). After plotting the vertex, choose additional x-values and plug them into the function to get corresponding y-values to plot other points. Because the coefficient \(a = -1\), the parabola opens downward.

Step 2: Finding the Axis of Symmetry

The axis of symmetry for a parabola in vertex form is the line \(x = h\), where \(h\) is the x-coordinate of the vertex. Since the vertex of the given function is at \( (7, 10) \), the axis of symmetry is the line \(x = 7\).

Key Concepts

Axis of Symmetry

The axis of symmetry in the context of graphing quadratic functions is a vertical line that divides the parabola into two symmetrical halves. It's like a mirror placed along the parabola, reflecting one side onto the other. The equation for the axis of symmetry can be straightforwardly determined when the quadratic function is in vertex form, which looks like y = a(x - h)^2 + k. In this form, the h value represents the x-coordinate of the vertex, and therefore gives us the equation for the axis of symmetry as x = h.

For instance, given the function y = -(x - 7)^2 + 10, the vertex form shows us that h = 7. Hence, the axis of symmetry is the line x = 7. This is essential in graphing because once you've plotted the vertex, you can use this line as a guide to place other points symmetrically.

Vertex of a Parabola

The vertex of a parabola represents the 'turning point' where the function switches direction. For a parabola that opens upwards, the vertex is the lowest point, and for one that opens downwards, the vertex is the highest point. When dealing with the vertex form of a quadratic, y = a(x - h)^2 + k, finding the vertex is a breeze. The coordinates (h, k) can be easily spotted within the equation itself, signaling the vertex's location on the graph.

In our example, y = -(x - 7)^2 + 10, the numbers 7 and 10 reveal that the vertex is at (7, 10). Knowing this, we understand that the parabola reaches its peak at this point — since the 'a' value is negative, indicating that the parabola opens downward — and every other point on the graph is mirrored across the axis of symmetry.

Vertex Form of a Parabola

The vertex form of a parabola is perhaps the most instructive when it comes to graphing. It's written as y = a(x - h)^2 + k, where a determines the 'width' and the direction of the opening, h and k are the coordinates of the vertex. In simple terms:

• If a is positive, the parabola opens upwards.
• If a is negative, the parabola opens downwards.
• If |a| is greater than 1, the parabola is 'narrower'.
• If |a| is less than 1, the parabola is 'wider'.

In our example, y = -(x - 7)^2 + 10 shows a parabola that opens downwards because a = -1. The vertex here is (7, 10), providing a central point for graphing. This form is incredibly user-friendly as it allows for easy identification of the parabola's characteristics, enabling students to graph with more confidence and understanding.