Table of contents
- 0. Review of Algebra4h 16m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 6m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 19m
- 10. Combinatorics & Probability1h 45m
4. Polynomial Functions
Quadratic Functions
Problem 25
Textbook Question
In Exercises 17–38, 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
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1
Identify the standard form of the quadratic function: \( f(x) = a(x-h)^2 + k \). Here, \( f(x) = 4 - (x-1)^2 \) can be rewritten as \( f(x) = -(x-1)^2 + 4 \).
Determine the vertex of the parabola. The vertex form \( f(x) = a(x-h)^2 + k \) indicates that the vertex is \((h, k)\). For this function, the vertex is \((1, 4)\).
Find the axis of symmetry. The axis of symmetry for a parabola in vertex form \( f(x) = a(x-h)^2 + k \) is the vertical line \( x = h \). Therefore, the axis of symmetry is \( x = 1 \).
Determine the x-intercepts by setting \( f(x) = 0 \) and solving for \( x \): \( 0 = 4 - (x-1)^2 \). Rearrange to find \( (x-1)^2 = 4 \), then solve for \( x \).
Determine the domain and range of the function. The domain of any quadratic function is all real numbers, \( (-\infty, \infty) \). The range is determined by the vertex and the direction of the parabola. Since the parabola opens downwards (as indicated by the negative coefficient of \( (x-1)^2 \)), the range is \( (-\infty, 4] \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vertex of a Quadratic Function
The vertex of a quadratic function is the highest or lowest point on the graph, depending on the direction of the parabola. For the function f(x) = 4 - (x - 1)², the vertex can be found by identifying the values of x and y at which the function reaches its maximum or minimum. In this case, the vertex is at the point (1, 4).
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Axis of Symmetry
The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. For a quadratic function in the form f(x) = a(x - h)² + k, the axis of symmetry is given by the line x = h. In the function f(x) = 4 - (x - 1)², the axis of symmetry is x = 1.
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Domain and Range of Quadratic Functions
The domain of a quadratic function is the set of all possible input values (x-values), which is typically all real numbers for parabolas. The range is the set of possible output values (y-values), which depends on the vertex. For f(x) = 4 - (x - 1)², the range is y ≤ 4, as the vertex represents the maximum point of the parabola.
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