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Chapter 3: Problem 25

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. $$f(x)=4-(x-1)^{2}$$

Short Answer

Expert verified
The vertex is at (1, 4), the x-intercepts are at (1, 0) and (3, 0), and the y-intercept is at (0, 3). Hence the axis of symmetry is \(x = 1\). The domain of the function is \(-\infty < x < \infty\) (all real numbers) and the range is \(-\infty < y \leq 4\).

Step by step solution

01

Calculate Vertex

The vertex of a quadratic equation in the standard form \(a(x-h)^2 + k\) is at the point \((h, k)\). For the function \(f(x) = 4-(x-1)^2\), the vertex of the parabola is at point \((1, 4)\).
02

Calculate Intercepts

The x-intercepts can be found by setting \(f(x)\) to 0 and solving for \(x\). Doing so for the equation \(4-(x-1)^2 = 0\) gives two x-intercepts at points \((1, 0)\) and \((3, 0)\). The y-intercept is found by substituting \(x\) with 0 in the function, that gives the single point \((0, 3)\).
03

Sketch the graph

Plot the vertex and intercepts on a graph. Then, draw a curve linking these points to form the graph of the quadratic function. Make sure the curve opens downwards as the coefficient of \(x^2\) is negative.
04

Determine the Axis of Symmetry

The axis of symmetry of the function is a vertical line through the vertex. So, for the vertex \((1, 4)\), the axis of symmetry is the line \(x = 1\).
05

Analyze the Domain and Range

From the graph, we can observe that the parabola, being symmetric about \(x = 1\) and opening downwards, extends infinitely towards both left and right directions and has a maximum value at y = 4. So the domain of the function is all real numbers or \(-\infty < x < \infty\), and its range is all y-values less than or equal to 4; i.e., \(-\infty < y \leq 4\).

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