Graph the function $y=\frac{1}{8}(x-4)^2+3$ on a coordinate plane using its vertex, focus, and directrix.
Answer (2)
Identify the vertex form of the parabola: y = a ( x − h ) 2 + k , where the vertex is ( h , k ) .
Determine the vertex from the given equation y = 8 1 ( x − 4 ) 2 + 3 , which is ( 4 , 3 ) .
Calculate the distance p from the vertex to the focus and directrix using a = 4 p 1 , which gives p = 2 .
Find the focus ( 4 , 5 ) and the directrix y = 1 , and graph the parabola using these elements. The final answer is vertex ( 4 , 3 ) , focus ( 4 , 5 ) , and directrix y = 1 .
Explanation
Problem Analysis We are given the equation of a parabola y = 8 1 ( x − 4 ) 2 + 3 and we need to graph it using its vertex, focus, and directrix. Let's first identify these key features.
Finding the Vertex The given equation is in the vertex form of a parabola, which is y = a ( x − h ) 2 + k , where ( h , k ) is the vertex of the parabola. Comparing the given equation y = 8 1 ( x − 4 ) 2 + 3 with the vertex form, we can identify the vertex as ( 4 , 3 ) .
Calculating the Distance p The coefficient a in the equation y = a ( x − h ) 2 + k determines whether the parabola opens upwards or downwards. In our case, a = 8 1 , which is positive, so the parabola opens upwards. The distance p from the vertex to the focus and from the vertex to the directrix is related to a by the formula a = 4 p 1 . Therefore, we can find p as follows: 8 1 = 4 p 1 4 p = 8 p = 2
Finding the Focus Since the parabola opens upwards, the focus is located above the vertex at a distance p . Thus, the coordinates of the focus are ( h , k + p ) = ( 4 , 3 + 2 ) = ( 4 , 5 ) .
Finding the Directrix The directrix is a horizontal line located below the vertex at a distance p . Thus, the equation of the directrix is y = k − p = 3 − 2 = 1 . So, the directrix is the line y = 1 .
Graphing the Parabola Now that we have the vertex ( 4 , 3 ) , the focus ( 4 , 5 ) , and the directrix y = 1 , we can graph the parabola. Plot the vertex, focus, and directrix on the coordinate plane. The parabola opens upwards, with the vertex as its lowest point. The parabola is symmetric with respect to the vertical line passing through the vertex (the axis of symmetry). To sketch the parabola, we can find a few additional points. For example, when x = 0 , y = 8 1 ( 0 − 4 ) 2 + 3 = 8 1 ( 16 ) + 3 = 2 + 3 = 5 . So, the point ( 0 , 5 ) is on the parabola. Due to symmetry, the point ( 8 , 5 ) is also on the parabola.
Final Answer In conclusion, the vertex of the parabola is ( 4 , 3 ) , the focus is ( 4 , 5 ) , and the directrix is y = 1 . These elements define the parabola y = 8 1 ( x − 4 ) 2 + 3 .
Examples
Parabolas have many real-world applications, such as in the design of satellite dishes and reflective telescopes. The focus of a parabolic reflector is the point where incoming parallel rays are concentrated, making it ideal for collecting signals or energy. Understanding the vertex, focus, and directrix of a parabola allows engineers to precisely design these devices for optimal performance. For example, knowing the focus helps in positioning the receiver in a satellite dish to capture the strongest signal. Similarly, parabolic mirrors in car headlights use the focus to direct light in a parallel beam, improving visibility.
The vertex of the parabola described by y = 8 1 ( x − 4 ) 2 + 3 is ( 4 , 3 ) , the focus is ( 4 , 5 ) , and the directrix is the line y = 1 . To graph it, plot these three points and draw the parabola opening upwards. The parabola is symmetric about the line x = 4 .
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