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Consider the function f(x) shown in the graph below. Which of the following are equations for f(x)? Select all that apply.

Consider The Function Fx Shown In The Graph Below Which Of The Following Are Equations For Fx Select All That Apply class=

Sagot :

The given graph has 1 and 3 as zeroes of the function, the graph is also parabolic downwards, so we know that the leading coefficient is negative. It's vertex is at (2,1)

[tex]\begin{gathered} f(x)=-(x-1)(x-3)\text{ based on the roots} \\ f(x)=-(x-2)^2+1\text{ based on the vertex} \\ f(x)=-x^2+4x-3\text{ if we simplify }-(-x-1)(x-3) \\ \end{gathered}[/tex][tex]\begin{gathered} \text{The standard form of a quadratic equation is }f(x)=a(x-h)^2+k \\ \text{where }(h,k)\text{ is the vertex} \end{gathered}[/tex]

By inspection of the graph, the vertex is found at (2,1), since it is facing downwards, it also gives us an idea that a is negative.

[tex]\begin{gathered} \text{The function} \\ f(x)=-(x-2)^2+1 \\ \text{is already on the the vertex form, and it tells us that }a\text{ is negative because it is preceeded} \\ \text{by a negative sign, and that the vertex is at }(2,1)\text{ which is found on the function.} \end{gathered}[/tex][tex]\begin{gathered} \text{The zeroes, or the x-intercept, are values of }x\text{ for which the function crosses the x-axis} \\ \text{Inspecting the graph, it crosses at }x=1,\text{ and }x=3 \\ \text{Equate these to zero and we get} \\ x=1 \\ x-1=0 \\ \\ x=3 \\ x-3=0 \\ \\ \text{which means that }(x-1)\text{ and }(x-3)\text{ are factors, and since it is facing downward it} \\ \text{will be preceeded by a negative sign hence} \\ f(x)=-(x-1)(x-3) \end{gathered}[/tex]

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