Problem 20
Question
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. \(f(x)=(x-3)^{2}+2\)
Step-by-Step Solution
Verified Answer
The vertex of the function \(f(x)=(x-3)^{2}+2\) is \(3, 2\). The axis of symmetry is \(x=3\). The x-intercepts are \(3\) and \(-1\) and it does not have a y-intercept. The domain of the function is all real numbers or \(-\infty < x < \infty\), and the range is \([2, \infty)\).
1Step 1: Find the Vertex
The function \(f(x)=(x-3)^{2}+2\) is in the vertex form where the vertex is \(h, k\). Therefore, the vertex is located at \(3, 2\).
2Step 2: Find the Axis of Symmetry
The axis of symmetry for a parabola is vertical line that goes through the vertex. So, in this case, the axis of symmetry is \(x=3\).
3Step 3: Sketch the Graph
The vertex is at \(3, 2\) and the axis of symmetry is \(x=3\). Plot the vertex on the graph. Since the coefficient of \(x^{2}\) is positive, the parabola opens upward. As a result, you can draw the parabola symmetrically on both sides of the axis of symmetry.
4Step 4: Determine the Intercepts
Set \(f(x)=0\) to find the x-intercepts, that is \((x-3)^{2}+2 = 0\). Solving this will give us \(x = 3, -1\). It's also clear that y-intercept doesn't exist since function doesn't touch y-axis.
5Step 5: Determine the Domain and Range
For any quadratic function, the domain is all real numbers because the function is defined for every value of x. So, here, domain is \(-\infty < x < \infty\). The range is the set of all y-values the function can take. For this function, since it opens upward and the vertex is \(2\), the range is \([2, \infty)\).
Key Concepts
Vertex of a ParabolaAxis of SymmetryQuadratic Function Domain and Range
Vertex of a Parabola
Understanding the vertex of a parabola is crucial when you're working with quadratic functions. A vertex is the highest or lowest point on a parabola, which is a U-shaped graph.
For the quadratic function given by the equation \(f(x)=(x-3)^{2}+2\), the equation is already in the vertex form \(f(x)=a(x-h)^2+k\), where \((h, k)\) is the vertex of the parabola. In this case, the vertex is at \((3, 2)\). This point is where the parabola turns and is also a key factor in determining the graph's shape. A helpful tip is that the parabola opens upwards if \(a\) is positive, and downwards if \(a\) is negative. In our exercise, the coefficient of \(x^2\) is positive, signifying a parabola that opens upward, and therefore, the vertex is at the minimum point on the graph.
For the quadratic function given by the equation \(f(x)=(x-3)^{2}+2\), the equation is already in the vertex form \(f(x)=a(x-h)^2+k\), where \((h, k)\) is the vertex of the parabola. In this case, the vertex is at \((3, 2)\). This point is where the parabola turns and is also a key factor in determining the graph's shape. A helpful tip is that the parabola opens upwards if \(a\) is positive, and downwards if \(a\) is negative. In our exercise, the coefficient of \(x^2\) is positive, signifying a parabola that opens upward, and therefore, the vertex is at the minimum point on the graph.
Axis of Symmetry
The axis of symmetry is a straight line that vertically splits the parabola into two mirror images. It passes through the vertex and gives the parabola its symmetrical nature. For the equation \(f(x)=(x-3)^{2}+2\), the axis of symmetry is represented by \(x=h\), where \(h\) is the x-coordinate of the vertex.
In our example, since the vertex is \((3, 2)\), the axis of symmetry is the vertical line \(x=3\). When sketching the graph of a quadratic function, the axis of symmetry can be a valuable reference line that ensures accuracy and symmetry. Once you've plotted the vertex, you can use the axis of symmetry to reflect any points on one side of the parabola over to the other.
In our example, since the vertex is \((3, 2)\), the axis of symmetry is the vertical line \(x=3\). When sketching the graph of a quadratic function, the axis of symmetry can be a valuable reference line that ensures accuracy and symmetry. Once you've plotted the vertex, you can use the axis of symmetry to reflect any points on one side of the parabola over to the other.
Quadratic Function Domain and Range
The domain and range of a quadratic function are essential characteristics that define where the function exists and what values it can output. The domain refers to all possible x-values that can be input into the function, while the range is the set of all possible y-values that the function can output.
In general, for any quadratic function, the domain will always be all real numbers because there are no restrictions on the x-values. Hence, the domain is \(-\infty < x < \infty\). Now, the range depends on the direction in which the parabola opens. For our function \(f(x)=(x-3)^{2}+2\), since it opens upward and the vertex is at the point \((3, 2)\), the y-value 2 is the lowest point on the graph. Consequently, the range starts from this minimum y-value and extends to infinity, which can be expressed as \([2, \infty)\). It's also worth noting that when a parabola opens downwards, the range would extend from negative infinity to the y-value of the vertex, which would be the highest point on the graph.
In general, for any quadratic function, the domain will always be all real numbers because there are no restrictions on the x-values. Hence, the domain is \(-\infty < x < \infty\). Now, the range depends on the direction in which the parabola opens. For our function \(f(x)=(x-3)^{2}+2\), since it opens upward and the vertex is at the point \((3, 2)\), the y-value 2 is the lowest point on the graph. Consequently, the range starts from this minimum y-value and extends to infinity, which can be expressed as \([2, \infty)\). It's also worth noting that when a parabola opens downwards, the range would extend from negative infinity to the y-value of the vertex, which would be the highest point on the graph.
Other exercises in this chapter
Problem 20
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Determine the constant of variation for each stated condition. \(a\) varies jointly as \(b\) and \(c,\) and \(a=72\) when \(b=18\) and \(c=2\)
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