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Chapter 8: Problem 19

(Graphing program required.) Given \(f(x)=-x^{2}+8 x-15\) : a. Estimate by graphing: the \(x\) -intercepts, the \(y\) -intercept, and the vertex. b. Calculate the coordinates of the vertex.

Short Answer

Expert verified
x-intercepts: estimated from graph, y-intercept: -15, vertex: (4, 1), vertex coordinates: (4, 1)

Step by step solution

01

- Graph the Function

Plot the function given by the equation \(f(x) = -x^2 + 8x - 15\) using a graphing program or graphing calculator. Observe the shape of the parabola.
02

- Identify the x-intercepts

Locate the points where the graph intersects the x-axis. These points are the x-intercepts. Estimate these points from the graph.
03

- Identify the y-intercept

Identify the point where the graph intersects the y-axis. This point is the y-intercept.
04

- Estimate the Vertex

Find the highest point on the graph of the parabola since the parabola opens downwards. This point is the vertex. Estimate the coordinates from the graph.
05

- Calculate the Vertex Coordinates

The vertex of a parabola given by the equation \(y = ax^2 + bx + c\) can be found using the formula \(x = \frac{-b}{2a}\). Substitute \(a = -1\), \(b = 8\), and \(c = -15\) to find the x-coordinate: \(x = \frac{-8}{2(-1)} = 4\). Substitute \(x = 4\) back into the equation to find the y-coordinate: \(y = -4^2 + 8(4) - 15 = -16 + 32 - 15 = 1\). So, the vertex is at \((4, 1)\).

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Key Concepts

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

x-intercepts
In any quadratic function of the form \( f(x) = ax^2 + bx + c \), the x-intercepts are the points where the graph crosses the x-axis. These are the values of \( x \) for which \( f(x) = 0 \). For the equation \( f(x) = -x^2 + 8x - 15 \), solving for the x-intercepts involves finding the roots of the quadratic equation.
  • First, set the equation equal to zero: \( -x^2 + 8x - 15 = 0 \).
  • Next, use the quadratic formula: \( x = \frac{-b \, \text{\textpm} \, \text{\textsqrt}{b^2 - 4ac}}{2a} \), where \( a = -1 \), \( b = 8 \), and \( c = -15 \).
  • Calculate the discriminant: \( b^2 - 4ac \). Here, it is \( 8^2 - 4(-1)(-15) = 64 - 60 = 4 \).
  • Finally, find the x-intercepts: \( x = \frac{-8 \, \text{\textpm} \, \text{\textsqrt}{4}}{2(-1)} \). The solutions are \( x = 3 \) and \( x = 5 \).
Thus, the parabola intersects the x-axis at \( x = 3 \) and \( x = 5 \).
y-intercept
The y-intercept of a quadratic function is the point where the graph crosses the y-axis. This occurs when \( x = 0 \). To find the y-intercept for the equation \( f(x) = -x^2 + 8x - 15 \), simply substitute \( x = 0 \) into the equation.
  • Set \( x = 0 \) in the equation: \( f(0) = -0^2 + 8(0) - 15 \).
  • Simplify the equation: \( f(0) = -15 \).
So, the y-intercept of the given function is \( (0, -15) \).
vertex calculation
The vertex of a parabola described by the quadratic function \( f(x) = ax^2 + bx + c \) represents its highest or lowest point, depending on the direction it opens. For the function \( f(x) = -x^2 + 8x - 15 \), the vertex can be found using the vertex formula, \( x = \frac{-b}{2a} \).
  • Identify the coefficients: \( a = -1 \) and \( b = 8 \).
  • Calculate the x-coordinate of the vertex: \( x = \frac{-8}{2(-1)} = 4 \).
Once the x-coordinate is found, substitute it back into the original equation to find the y-coordinate:
  • Substitute \( x = 4 \) in \( f(x) \): \( f(4) = -4^2 + 8(4) - 15 \).
  • Simplify the expression: \( f(4) = -16 + 32 - 15 = 1 \).
Therefore, the vertex of the parabola is located at \( (4, 1) \).
graphing quadratic functions
Graphing a quadratic function involves understanding the shape and key features of its parabola. The general form is \( f(x) = ax^2 + bx + c \). Here's how to graph the function accurately:
  • Identify the direction: If \( a \) is positive, the parabola opens upwards. If \( a \) is negative, it opens downwards. For \( f(x) = -x^2 + 8x - 15 \), \( a = -1 \), so the parabola opens downwards.
  • Locate the key points: These include the x-intercepts, y-intercept, and vertex. Use the calculations described earlier. For our function, we have:
    • x-intercepts: \( (3, 0) \) and \( (5, 0) \)
    • y-intercept: \( (0, -15) \)
    • vertex: \( (4, 1) \)
  • Plot the points: Draw the x-intercepts, y-intercept, and vertex on a graph.
  • Draw the parabola: Connect these points smoothly to form a U-shaped curve (in this case, an upside-down U due to the negative \( a \)). Ensure the vertex is the highest point, and the ends extend both leftwards and rightwards infinitely.
  • Check your work: Make sure all calculated points and the direction of the parabola match the original quadratic function.
Following these steps ensures an accurate graph representation of the quadratic function.

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Most popular questions from this chapter

(Graphing program optional) a. Write each of the following functions in both the \(a-b-c\) and the \(a-h-k\) forms. Is one form easier than the other for finding the vertex? The \(x\) - and \(y\) -intercepts? \(y_{1}=2 x^{2}-3 x-20 \quad y_{3}=3 x^{2}+6 x+3\) \(y_{2}=-2(x-1)^{2}-3 \quad y_{4}=-(2 x+4)(x-3)\) b. Find the vertex and \(x\) - and \(y\) -intercepts and construct a graph by hand for each function in part (a). If you have access to a graphing program, check your work.

Put each of the following quadratics into standard form. a. \(f(x)=(x+3)(x-1)\) b. \(P(t)=(t-5)(t+2)\) c. \(H(z)=(2+z)(1-z)\)

Calculate the coordinates of the \(x\) - and \(y\) -intercepts for the following quadratics. a. \(y=3 x^{2}+2 x-1\) c. \(y=(5-2 x)(3+5 x)\) b. \(y=3(x-2)^{2}-1\) d. \(f(x)=x^{2}-5\)

The management of a company is negotiating with a union over salary increases for the company's employees for the next 5 years. One plan under consideration gives each worker a bonus of \(\$ 1500\) per year. The company currently employs 1025 workers and pays them an average salary of \(\$ 30,000\) a year. It also plans to increase its workforce by 20 workers a year. a. Construct a function \(C(t)\) that models the projected cost of this plan (in dollars) as a function of time \(t\) (in years). b. What will the annual cost be in 5 years?

(Graphing program required.) At low speeds an automobile engine is not at its peak efficiency; efficiency initially rises with speed and then declines at higher speeds. When efficiency is at its maximum, the consumption rate of gas (measured in gallons per hour) is at a minimum. The gas consumption rate of a particular car can be modeled by the following equation, where \(G\) is the gas consumption rate in gallons per hour and \(M\) is speed in miles per hour: \(G=0.0002 M^{2}-0.013 M+1.07\) a. Construct a graph of gas consumption rate versus speed. Estimate the minimum gas consumption rate from your graph and the speed at which it occurs. b. Using the equation for \(G,\) calculate the speed at which the gas consumption rate is at its minimum. What is the minimum gas consumption rate? c. If you travel for 2 hours at peak efficiency, how much gas will you use and how far will you go? d. If you travel at \(60 \mathrm{mph}\), what is your gas consumption rate? How long does it take to go the same distance that you calculated in part (c)? (Recall that travel distance = speed \(\times\) time traveled.) How much gas is required for the trip? e. Compare the answers for parts (c) and (d), which tell you how much gas is used for the same-length trip at two different speeds. Is gas actually saved for the trip by traveling at the speed that gives the minimum gas consumption rate? f. Using the function \(G,\) generate data for gas consumption rate measured in gallons per mile by completing the following table. Plot gallons per mile (on the vertical axis) vs. miles per hour (on the horizontal axis). At what speed is gallons per mile at a minimum? g. Add a fourth column to the data table. This time compute miles/gal \(=\mathrm{mph} /(\mathrm{gal} / \mathrm{hr}) .\) Plot miles per gallon vs. miles per hour. At what speed is miles per gallon at a maximum? This is the inverse of the preceding question; we are normally used to maximizing miles per gallon instead of minimizing gallons per mile. Does your answer make sense in terms of what you found for parts (b) and (f)?
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