Problem 22
Question
Describe the graph of the function and identify the vertex. Use a graphing utility to verify your results. \(f(x)=(x-7)^{2}+2\)
Step-by-Step Solution
Verified Answer
The graph of the function \(f(x) = (x-7)^{2}+2\) is an upward opening parabola with its vertex at (7,2).
1Step 1: Identify the Form of the Function
The given function, \(f(x)=(x-7)^{2}+2\), is in the form \(f(x) = a(x-h)^{2}+k\), where \(a\), \(h\), and \(k\) are constants. It's known that this is a format of a quadratic function, with the vertex being \((h,k)\).
2Step 2: Find the vertex
Compare the given function with the general form to find the vertex: \(h = 7\) and \(k = 2\). Hence, the vertex \((h,k)\) of the parabola is \((7,2)\).
3Step 3: Determine the shape and direction
Since \(a\) in \(f(x) = a(x-h)^{2}+k\) is positive and equal to 1, the graph opens upward. The minimum point on the graph will be the vertex.
4Step 4: Verify your results with a graphing utility
Using any graphing utility, plot the given function \(f(x)=(x-7)^{2}+2\). It can be confirmed that the graph is a parabola that opens upward and the vertex point is (7,2), confirming the initially deduced values.
Key Concepts
Vertex of a ParabolaGraphing Quadratic FunctionsStandard Form of a Quadratic Equation
Vertex of a Parabola
Understanding the vertex of a parabola is essential when dealing with quadratic functions. The vertex represents the highest or lowest point of the parabola, depending on whether the parabola opens upwards or downwards. For the quadratic function given by the general form
\( f(x) = a(x-h)^2 + k \), the vertex is at the point \((h, k)\). In this form, \(h\) and \(k\) are easily identified as the coordinates of the vertex, making it straightforward to determine the parabola's peak or trough.
In the exercise provided, the function is \(f(x) = (x-7)^2 + 2\), where we can immediately see that the vertex is at \((7, 2)\). The vertex is crucial because it not only helps in sketching the graph accurately but also provides valuable information about the function's maximum or minimum value, and by extension, insights into the real-world scenarios modeled by such functions. When graphing by hand, you would plot the vertex first and then use additional points to define the curvature of the parabola.
\( f(x) = a(x-h)^2 + k \), the vertex is at the point \((h, k)\). In this form, \(h\) and \(k\) are easily identified as the coordinates of the vertex, making it straightforward to determine the parabola's peak or trough.
In the exercise provided, the function is \(f(x) = (x-7)^2 + 2\), where we can immediately see that the vertex is at \((7, 2)\). The vertex is crucial because it not only helps in sketching the graph accurately but also provides valuable information about the function's maximum or minimum value, and by extension, insights into the real-world scenarios modeled by such functions. When graphing by hand, you would plot the vertex first and then use additional points to define the curvature of the parabola.
Graphing Quadratic Functions
Graphing quadratic functions involves plotting a parabolic curve on a coordinate plane. Quadratic functions are typically represented in the standard form \( ax^2 + bx + c \), vertex form \( a(x-h)^2 + k \), or factored form. The process of graphing begins by identifying key features such as the vertex, axis of symmetry, direction of the opening (upward or downward), and the y-intercept.
For the function \(f(x) = (x-7)^2 + 2\), the vertex \((7, 2)\) provides a starting point. We know the parabola opens upwards because the coefficient \(a\) is positive. The axis of symmetry is the vertical line that passes through the vertex, which, in this case, is \(x = 7\). By plotting additional points on either side of the axis of symmetry and connecting them in a smooth, continuous curve, the shape of the parabola becomes apparent. Graphing utilities or software can aid in this process by providing a precise and accurate representation of the quadratic function.
For the function \(f(x) = (x-7)^2 + 2\), the vertex \((7, 2)\) provides a starting point. We know the parabola opens upwards because the coefficient \(a\) is positive. The axis of symmetry is the vertical line that passes through the vertex, which, in this case, is \(x = 7\). By plotting additional points on either side of the axis of symmetry and connecting them in a smooth, continuous curve, the shape of the parabola becomes apparent. Graphing utilities or software can aid in this process by providing a precise and accurate representation of the quadratic function.
Standard Form of a Quadratic Equation
The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) are coefficients and \(x\) represents the variable. This form is particularly useful for analyzing the components of a quadratic function and is the starting point for various methods of solving quadratics, such as factoring, completing the square, and using the quadratic formula.
However, when graphing, it is often more convenient to work with the vertex form of the equation, \( a(x-h)^2 + k \), because it readily displays the vertex. In the given exercise, we already have the equation in vertex form, which allows us to easily extract the vertex. To convert from the standard form to the vertex form, one would typically complete the square, a process which involves creating a perfect square trinomial. Appreciating the standard form's role in the broader context of quadratic functions ensures that students can navigate between different forms and recognize their unique advantages in various mathematical tasks.
However, when graphing, it is often more convenient to work with the vertex form of the equation, \( a(x-h)^2 + k \), because it readily displays the vertex. In the given exercise, we already have the equation in vertex form, which allows us to easily extract the vertex. To convert from the standard form to the vertex form, one would typically complete the square, a process which involves creating a perfect square trinomial. Appreciating the standard form's role in the broader context of quadratic functions ensures that students can navigate between different forms and recognize their unique advantages in various mathematical tasks.
Other exercises in this chapter
Problem 22
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Find all the zeros of the function and write the polynomial as a product of linear factors. Use a graphing utility to verify your results graphically. (If possi
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Sketch the graph of \(f(x)=x^{3}\) and the graph of the function \(g .\) Describe the transformation from \(f\) to \(g .\) \(g(x)=(x+4)^{3}+1\)
View solution Problem 23
Find any asymptotes and holes in the graph of the rational function. Verify your answers by using a graphing utility. $$f(x)=\frac{x^{2}-16}{x^{2}+8 x}$$
View solution