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Chapter 3: Problem 26

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. $$f(x)=1-(x-3)^{2}$$

Short Answer

Expert verified
The axis of symmetry is \(x = 3\). The domain is all real numbers. The range is \(y \leq 1\).

Step by step solution

01

Plotting the Function

The function provided is \(f(x) = 1-(x-3)^{2}\). We can plot this function on a graph. First, it's important to recognize the vertex form of a quadratic function: \(f(x) = a(x-h)^{2} + k\), where (h, k) are the vertex coordinates. Hence, our vertex for this function is at (3, 1). We can plot further points by assigning different x-values and solving for y to get the corresponding y-values.
02

Determining the Axis of Symmetry

The axis of symmetry of a parabola described by a function \(f(x) = a(x-h)^{2} + k\) is the line \(x=h\). Thereby, the axis of symmetry for this function is \(x=3\). This line of symmetry can be drawn in the same graph where the function was plotted.
03

Determining the Domain and Range

Next, let's use the graph to determine the domain and range of the function. The domain of a function represents all possible x-values, and for quadratic functions, it is all real numbers, \(R\). The range of a function represents all possible y-values. Looking at the graph, we can see that the maximum y-value is 1 (the y-coordinate of the vertex). And as this is a downward-opening parabola (since \(a<0\)), the parabola will continue to decrease as we move away from the vertex on either side. Therefore, the range is \(y \leq 1\).

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