Problem 18
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. $$f(x)-(x-1)^{2}-2$$
Step-by-Step Solution
Verified Answer
The parabola's vertex is at (1, -2). It is a upward opening parabola with the axis of symmetry at \(x=1\). The domain of the function is all real numbers, or (-∞, ∞), and its range is \([-2, ∞)\).
1Step 1: Identify the Vertex
The vertex of a quadratic function given in the form \(f(x) = a(x - h)^2 + k\) is located at the point (h, k). In this case, the function is \(f(x) = (x-1)^2 - 2\), so the vertex is at the point (1, -2).
2Step 2: Sketch the Graph
Knowing the vertex helps in sketching the graph. Because this is a parabola that opens upwards (as the coefficient of \(x^2\) is positive), the graph will be a U shaped graph with the lowest point at the vertex (1, -2). The graph will symmetric around the line \(x=h\).
3Step 3: Determine the Axis of Symmetry
The axis of symmetry of a parabola given in the form \(f(x) = a(x - h)^2 + k\) is the vertical line x=h. So in this case, the axis of symmetry will be the line \(x=1\).
4Step 4: Determine the Domain and Range
The domain of a function is the set of all real x-values for which the function is defined, and for any quadratic function, this is all real numbers, i.e., (-∞, ∞). The range of a function is the set of all real y-values that the function can take. Since the graph of this function opens upwards and its vertex is at y=-2, this means that the function can take any y-value greater than or equal to -2. Hence, the range is \([-2, ∞)\).
Key Concepts
Vertex of a Quadratic FunctionAxis of Symmetry of a ParabolaDomain and Range of a Function
Vertex of a Quadratic Function
Understanding the vertex of a quadratic function is key to graphing and analyzing the function's properties. The vertex is the point where the parabola either reaches its maximum or minimum value, depending on whether it opens upwards or downwards.
In the standard form, a quadratic function is expressed as \(f(x) = a(x - h)^2 + k\), where \((h, k)\) represents the vertex of the parabola. If the value of \(a\) is positive, the parabola opens upwards and the vertex represents the lowest point on the graph. Conversely, if \(a\) is negative, the parabola opens downwards and the vertex is the highest point.
For instance, in the given function \(f(x) = (x-1)^2 - 2\), we can see that \(a\) is positive, meaning the parabola opens upwards and the vertex, located at \((1, -2)\), is the lowest point of the graph. This point is crucial as it helps to determine the parabola's axis of symmetry and assists in finding the range of the function.
In the standard form, a quadratic function is expressed as \(f(x) = a(x - h)^2 + k\), where \((h, k)\) represents the vertex of the parabola. If the value of \(a\) is positive, the parabola opens upwards and the vertex represents the lowest point on the graph. Conversely, if \(a\) is negative, the parabola opens downwards and the vertex is the highest point.
For instance, in the given function \(f(x) = (x-1)^2 - 2\), we can see that \(a\) is positive, meaning the parabola opens upwards and the vertex, located at \((1, -2)\), is the lowest point of the graph. This point is crucial as it helps to determine the parabola's axis of symmetry and assists in finding the range of the function.
Axis of Symmetry of a Parabola
The axis of symmetry of a parabola is a vertical line that divides the parabola into two mirror-image halves. It is always a straight line passing through the vertex, hence it holds the equation \(x=h\) where \(h\) is the x-coordinate of the vertex.
In the context of the quadratic function \(f(x) = (x-1)^2 - 2\), the vertex is at \((1, -2)\). Accordingly, the axis of symmetry is the line \(x=1\). This line is not just a theoretical concept; it is used practically to ensure the graph is accurately sketched by reflecting points on one side of the parabola to the other.
Further, the axis of symmetry is essential in solving problems related to quadratics, such as finding the maximum or minimum value of the function, as this value lies on the axis of symmetry.
In the context of the quadratic function \(f(x) = (x-1)^2 - 2\), the vertex is at \((1, -2)\). Accordingly, the axis of symmetry is the line \(x=1\). This line is not just a theoretical concept; it is used practically to ensure the graph is accurately sketched by reflecting points on one side of the parabola to the other.
Further, the axis of symmetry is essential in solving problems related to quadratics, such as finding the maximum or minimum value of the function, as this value lies on the axis of symmetry.
Domain and Range of a Function
The domain and range of a function are fundamental concepts in understanding how a function behaves across the set of all possible inputs and outputs.
The domain includes all the possible x-values that a function can accept. For any quadratic function, such as \(f(x) = (x-1)^2 - 2\), the domain is all real numbers, expressed as \((-\infty, \infty)\). This is because you can input any x-value into a quadratic equation and get a corresponding y-value.
The range, on the other hand, consists of all the y-values that the function can output. Since the graph of our example opens upwards and the vertex is \((1, -2)\), it means the y-value starts at -2 and goes to positive infinity. Thus, the range of this function is \([-2, \infty)\), indicating that -2 is the minimum y-value (inclusive), and there is no upper limit to the y-values that the function can produce.
Understanding these concepts enables students to predict and describe the behavior of quadratic functions, and importantly, to sketch their graphs accurately.
The domain includes all the possible x-values that a function can accept. For any quadratic function, such as \(f(x) = (x-1)^2 - 2\), the domain is all real numbers, expressed as \((-\infty, \infty)\). This is because you can input any x-value into a quadratic equation and get a corresponding y-value.
The range, on the other hand, consists of all the y-values that the function can output. Since the graph of our example opens upwards and the vertex is \((1, -2)\), it means the y-value starts at -2 and goes to positive infinity. Thus, the range of this function is \([-2, \infty)\), indicating that -2 is the minimum y-value (inclusive), and there is no upper limit to the y-values that the function can produce.
Understanding these concepts enables students to predict and describe the behavior of quadratic functions, and importantly, to sketch their graphs accurately.
Other exercises in this chapter
Problem 18
In Exercises \(17-24\) a. List all possible rational roots. b. Use synthetic division to test the possible rational roots and find an actual root. c. Use the qu
View solution Problem 18
Divide using synthetic division. $$ \left(x^{2}+x-2\right) \div(x-1) $$
View solution Problem 19
Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ x^{2
View solution Problem 19
Write an equation that expresses each relationship. Then solve the equation for y. \(x\) varies directly as \(z\) and inversely as the difference between \(y\)
View solution