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Chapter 3: Problem 9

For each quadratic function, (a) write the function in the form \(P(x)=a(x-h)^{2}+k,\) (b) give the vertex of the parabola, and (c) graph the function. Do not use a calculator. $$P(x)=x^{2}-6 x$$

Short Answer

Expert verified
The function is \(P(x) = (x-3)^2 - 9\) with vertex (3, -9).

Step by step solution

01

Completing the Square

Given the function \(P(x) = x^2 - 6x\), we first complete the square to rewrite it in the form \(P(x) = a(x-h)^2 + k\). We take half of the coefficient of \(x\), which is \(-6\), divide it by 2 to get \(-3\), and then square this to get \(9\). Thus, we have:\(P(x) = x^2 - 6x + 9 - 9\).This can be rewritten as:\(P(x) = (x - 3)^2 - 9\).The function is now in the form \(P(x) = a(x-h)^2 + k\), where \(a = 1\), \(h = 3\), and \(k = -9\).
02

Finding the Vertex

The vertex form of a quadratic is \(P(x) = a(x-h)^2 + k\), where \((h, k)\) is the vertex. From Step 1, we found \(P(x) = (x-3)^2 - 9\). Thus, the vertex of the parabola is \((3, -9)\).
03

Graphing the Function

To graph \(P(x) = (x-3)^2 - 9\), start by plotting the vertex at the point \((3, -9)\). Since the coefficient \(a = 1\) is positive, the parabola opens upwards. Plot additional points to the left and right of the vertex to ensure the shape, such as \((2, -8)\) and \((4, -8)\), verifying symmetry. Connect these points to form the parabola.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Completing the Square
Completing the square is a technique used to transform a standard quadratic equation into a form that is easily analyzed. This is useful in graphing and finding key features like the vertex of the parabola. Take the quadratic function \(P(x) = x^2 - 6x\). Begin by identifying the coefficient of the linear term, here it is \(-6\). Take half of this coefficient, which is \(-3\), and square it to get \(9\).
This value, \(9\), is added and subtracted within the function to allow restructuring. Thus, the function \(P(x) = x^2 - 6x\) becomes \(P(x) = x^2 - 6x + 9 - 9\), simplifying to \(P(x) = (x - 3)^2 - 9\).
The function is now expressed in vertex form \(P(x) = a(x-h)^2 + k\), where \(a = 1\), \(h = 3\), and \(k = -9\). This method effectively isolates the perfect square trinomial, simplifying further analysis.
Vertex Form
The vertex form of a quadratic function is \(P(x) = a(x-h)^2 + k\). This form is particularly advantageous because it quickly reveals the vertex of the parabola, an essential feature for graphing. In our example, after completing the square, we have \(P(x) = (x - 3)^2 - 9\).
The vertex \((h, k)\) directly emerges as \((3, -9)\). The vertex tells us the highest or lowest point on the graph depending on the direction of the parabola. Here, since \(a = 1\) (positive), it opens upwards.
  • \((h, k)\) indicates the turning point of the parabola.
  • Knowing \(a\) helps determine the direction of the parabola's opening.
Recognizing the vertex form enables quick identification and graphing without further complex calculations.
Graphing Quadratics
Graphing a quadratic function once in vertex form is straightforward and visual. Begin by locating the vertex of the parabola. For \(P(x) = (x-3)^2 - 9\), the vertex is at \((3, -9)\). This becomes your first point.
Next, note the value of \(a\) (in our case, \(a = 1\)), which dictates the direction of the parabola's opening. A positive \(a\) implies an upward opening. After plotting the vertex, choose symmetrical points around this vertex to maintain the parabola's shape. For instance, when \(x = 2\), \(P(x) = -8\), and similarly for \(x = 4\).
  • Vertex: Plot the vertex point first.
  • Symmetrical Points: Select points equidistant from the vertex for accurate shape.
  • Connected Curve: Draw a smooth curve through the points to complete the parabola.
This method ensures your graph reflects the quadratic's properties accurately.

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Most popular questions from this chapter

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(2 x+1 \geq x^{2}\) (b) \(2 x+1

Evaluate the discriminant, and use it to determine the number of real solutions of the equation. If the equation does have real solutions, tell whether they are rational or irrational. Do not actually solve the equation. $$2 x^{2}-4 x+1=0$$

Solve each problem. The coast-down time \(y\) for a typical car as it drops \(10 \mathrm{mph}\) from an initial speed \(x\) depends on several factors, such as average drag, tire pressure, and whether the transmission is in neutral. The table gives the coast-down time in seconds for a car under standard conditions for selected speeds in miles per hour. $$\begin{array}{|c|c|} \hline \begin{array}{c} \text { Initial Speed } \\ \text { (in mph) } \end{array} & \begin{array}{c} \text { Coast-Down } \\ \text { Time (in seconds) } \end{array} \\ \hline 30 & 30 \\ 35 & 27 \\ 40 & 23 \\ 45 & 21 \\ 50 & 18 \\ 55 & 16 \\ 60 & 15 \\ 65 & 13 \\ \hline \end{array}$$ (a) Plot the data. (b) Use the quadratic regression feature of a graphing calculator to find the quadratic function \(g\) that best fits the data. Graph this function in the same window as the data. Is \(g\) a good model for the data? (c) Use \(g\) to predict the coast-down time, to the nearest second, at an initial speed of 70 mph. (d) Use the graph to find the speed that comesponds to a coast-down time of 24 seconds.

Solve each equation for \(x\) and then for \(y .\) $$x^{2}+x y+y^{2}=0 \quad(x>0, y>0)$$

Evaluate the discriminant, and use it to determine the number of real solutions of the equation. If the equation does have real solutions, tell whether they are rational or irrational. Do not actually solve the equation. $$x^{2}+8 x+16=0$$
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