This tells us the paper will lose 2,500 subscribers for each dollar they raise the price. Treating C as the input and S as the output, the equation will have form. From this we can find a linear equation relating the two quantities. We know that currently S = 84,000 and C = 30, and that if they raise the price to $32 they would lose 5,000 subscribers, giving a second pair of values, C = 32 and S = 79,000. Since the number of subscribers changes with the price, we need to find a relationship between the variables. We can introduce variables, C for charge per subscription and S for the number subscribers, giving us the equation: In this case, the revenue can be found by multiplying the charge per subscription times the number of subscribers. Revenue is the amount of money a company brings in. Assuming that subscriptions are linearly related to the price, what price should the newspaper charge for a quarterly subscription to maximize their revenue? Market research has suggested that if they raised the price to $32, they would lose 5,000 subscribers. To maximize the area, she should enclose the garden so the two shorter sides have length 20 feet, and the longer side parallel to the existing fence has length 40 feet.ī.A local newspaper currently has 84,000 subscribers, at a quarterly charge of $30. When the shorter sides are 20 feet, that leaves 40 feet of fencing for the longer side. The maximum value of the function is an area of 800 square feet, which occurs when L = 20 feet. But we know that a is the coefficient on the squared term, so a = -2, b = 80, and c = 0. In finding the vertex, we take care since the equation is not written in standard polynomial form with decreasing powers. Notice that quadratic has been vertically reflected, since the coefficient on the squared term is negative, so the graph will open downwards, and the vertex will be a maximum value for the area. Returning to our backyard farmer from the beginning of the section, what dimensions should she make her garden to maximize the enclosed area?Įarlier we determined the area she could enclose with 80 feet of fencing on three sides was given by the equation. This same pattern can be used with any quadratic to quickly sketch the graph.Ī. Also notice that if we move from the vertex and go left or right 1 unit from the vertex, the point on the graph is 1/2 unit up and if you go another unit over, you go up 1 1\2 or 3 1/2 units up, which is units. Look back at the graph in the previous example. The vertex form of a quadratic is the simplest form to graph. When a < 0, the graph of the quadratic will open downwards. When a > 0, the graph of the quadratic will open upwards. This line is called the axis of symmetry. Graphs of quadratic functions are also symmetric around a vertical line through the vertex.The shape of the graph of a quadratic functions is called a parabola.The vertex of the quadratic function is located at, where h and k are the numbers in the vertex form of the function.The vertex form of a quadratic function is The standard form of a quadratic function is The standard form for a quadratic is, but you will also see them written in the form. In addition to intercepts, quadratics have an interesting feature where they change direction, called the vertex. We now explore the interesting features of the graphs of quadratics. This formula represents the area of the fence in terms of the variable length L. We know the area of a rectangle is length multiplied by width, so Now we are ready to write an equation for the area the fence encloses. This allows us to represent the width, W, in terms of L. Since we know we only have 80 feet of fence available, we know that, or more simply. It might also be helpful to introduce a temporary variable, W, to represent the side of fencing parallel to the 4th side or backyard fence. In a scenario like this involving geometry, it is often helpful to draw a picture. Find a formula for the area enclosed by the fence if the sides of fencing perpendicular to the existing fence have length L.
She has purchased 80 feet of wire fencing to enclose 3 sides, and will put the 4th side against the backyard fence. A backyard farmer wants to enclose a rectangular space for a new garden.
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