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Magazine Chapter 1: Problem 9
Generate the graph of \(f\) in a viewing window that you think is appropriate. $$f(x)=x^{2}-9 x-36$$
Short Answer
Expert verified
The function is a parabola with a vertex at (4.5, -56.25) and intercepts; it can be graphed in a window with x: [-5, 15] and y: [-60, 10].
Step by step solution
01
Identify the Function
The function given is a quadratic function of the form \(f(x) = ax^2 + bx + c\), where \(a = 1\), \(b = -9\), and \(c = -36\). For this function, the curve will be a parabola.
02
Determine the Vertex
The vertex form of a quadratic function is derived using the formula \(x = -\frac{b}{2a}\). Here, \(-\frac{-9}{2 \cdot 1} = 4.5\). Substitute back into the function to find the \(y\)-coordinate: \(f(4.5) = (4.5)^2 - 9 \cdot 4.5 - 36 = -56.25\). The vertex is at the point \((4.5, -56.25)\).
03
Find Intercepts
Find the \(y\)-intercept by setting \(x = 0\), giving \(f(0) = 0^2 - 9 \times 0 - 36 = -36\), so the \(y\)-intercept is \((0, -36)\). To find the \(x\)-intercepts, solve \(x^2 - 9x - 36 = 0\) using the quadratic formula: \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\). Here the discriminant \(=9^2-4\times1\times(-36)= 189\), so \(x = \frac{9\pm \sqrt{189}}{2}\).
04
Define Viewing Window
Considering the vertex at \((4.5, -56.25)\) and intercepts found above, choose a window that includes slightly beyond the intercepts and the vertex. A good range might be \(x: [-5, 15]\) and \(y: [-60, 10]\) to capture the full shape of the parabola.
05
Plot the Function
Plot the function over the defined range. A calculator or graphing tool is generally used, plotting key points like the vertex and intercepts to visualize the function. Confirm that the shape is indeed a parabola opening upwards, intersecting the \(x\)-axis as calculated.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Parabola
A parabola is a U-shaped curve that is the graph of a quadratic function. It is a distinctive shape that opens either upwards or downwards. The general form of a quadratic function is given by:
The sign of \(a\) determines the direction in which the parabola opens:
The parabola is symmetrical about a vertical line that passes through its vertex. Understanding this behavior helps in sketching and interpreting the graph of a quadratic function.
- \( f(x) = ax^2 + bx + c \)
The sign of \(a\) determines the direction in which the parabola opens:
- If \(a > 0\), the parabola opens upwards.
- If \(a < 0\), the parabola opens downwards.
The parabola is symmetrical about a vertical line that passes through its vertex. Understanding this behavior helps in sketching and interpreting the graph of a quadratic function.
Vertex of a Parabola
The vertex of a parabola is a crucial point that indicates its highest or lowest point, depending on its orientation. For an upward-opening parabola, like in our example \(f(x) = x^2 - 9x - 36\), the vertex is the minimum point.
To find the vertex, the formula \(x = -\frac{b}{2a}\) is used, where \(b\) and \(a\) are coefficients from the parabola's equation \( ax^2 + bx + c \).
In this instance:
This point is essential for graphing and helps to determine the parabola's axis of symmetry.
To find the vertex, the formula \(x = -\frac{b}{2a}\) is used, where \(b\) and \(a\) are coefficients from the parabola's equation \( ax^2 + bx + c \).
In this instance:
- \( x = -\frac{-9}{2 \times 1} = 4.5 \)
- \( f(4.5) = (4.5)^2 - 9 \times 4.5 - 36 = -56.25 \)
This point is essential for graphing and helps to determine the parabola's axis of symmetry.
Quadratic Formula
The Quadratic Formula is a vital tool used to find the roots or \(x\)-intercepts of a quadratic equation, which are the points where the parabola intersects the \(x\)-axis. It is given by:
First, compute the discriminant \(b^2 - 4ac\):
- \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
First, compute the discriminant \(b^2 - 4ac\):
- \( (-9)^2 - 4 \times 1 \times (-36) = 189 \)
- If it is positive, there are two distinct real roots.
- If it is zero, there is one real root (the parabola touches the \(x\)-axis at one point).
- If it is negative, there are no real roots (the parabola does not intersect the \(x\)-axis).
- \( x = \frac{9 \pm \sqrt{189}}{2} \)
Graphing Quadratics
Graphing quadratics involves plotting a parabola on the coordinate plane. The parabola's equation, key points like vertex and intercepts, and the quadratic formula, all help in plotting accurately.
For the function \(f(x) = x^2 - 9x - 36\), follow these steps to graph the parabola:
Finally, plot these points and draw a smooth curve through them to visualize the quadratic function graphically. Various tools, including graphing calculators or software, can make this process straightforward and accurate.
For the function \(f(x) = x^2 - 9x - 36\), follow these steps to graph the parabola:
- Identify key points: the vertex \((4.5, -56.25)\) and intercepts.
- Calculate the \(y\)-intercept by setting \(x = 0\), resulting in \(f(0) = -36\), giving the intercept \((0, -36)\).
- Use the quadratic formula to find \(x\)-intercepts or roots as described earlier.
Finally, plot these points and draw a smooth curve through them to visualize the quadratic function graphically. Various tools, including graphing calculators or software, can make this process straightforward and accurate.
