Which of the following correctly describes the graph of 𝑓(𝑥)=−4𝑥2+16𝑥−2 ?Group of answer choicesThe graph is an exponential function with a y-intercept at (2,14) and a horizontal asymptote at 𝑦=−2.The graph is a parabola with a vertex at (16,2), a y-intercept at (0,2), and the parabola opens down.The graph is a parabola with a vertex at (2,14), a y-intercept at (0,−2), and the parabola opens down.The graph is an exponential function with a y-intercept at (0,−2) and a horizontal asymptote at 𝑦=2.The graph is a parabola with a vertex at (2,14), a y-intercept at (0,2), and the parabola opens up.
Question
Which of the following correctly describes the graph of 𝑓(𝑥)=−4𝑥2+16𝑥−2 ?Group of answer choicesThe graph is an exponential function with a y-intercept at (2,14) and a horizontal asymptote at 𝑦=−2.The graph is a parabola with a vertex at (16,2), a y-intercept at (0,2), and the parabola opens down.The graph is a parabola with a vertex at (2,14), a y-intercept at (0,−2), and the parabola opens down.The graph is an exponential function with a y-intercept at (0,−2) and a horizontal asymptote at 𝑦=2.The graph is a parabola with a vertex at (2,14), a y-intercept at (0,2), and the parabola opens up.
Solution
The graph is a parabola with a vertex at (2,14), a y-intercept at (0,−2), and the parabola opens down.
Here's why:
The given function is 𝑓(𝑥)=−4𝑥²+16𝑥−2. This is a quadratic function, which means the graph will be a parabola.
The vertex form of a parabola is 𝑓(𝑥)=a(𝑥−h)²+k, where (h,k) is the vertex of the parabola.
We can find the vertex (h,k) by using the formula h = -b/2a and k = f(h). In this case, a = -4 and b = 16, so h = -16/(2*-4) = 2. Substituting x = 2 into the function gives us k = -42²+162-2 = 14. So, the vertex is at (2,14).
The y-intercept is found by setting x = 0 in the function, which gives us -2. So, the y-intercept is at (0,-2).
Finally, because the coefficient of 𝑥² is negative, the parabola opens down.
Similar Questions
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