From the graph we can infer that our function crosses the x-axis at -2.5, 0.5, and 1.5, so those are the zeroes of the function. Next, lets convert those decimals into fractions by adding a denominator 1 to each decimal and multiply both its numerator and denominator by a power 10 that lets us get rid of the decimal point: [tex]( \frac{-2.5}{1} )( \frac{10}{10} )= \frac{-25}{10} =- \frac{5}{2} [/tex] [tex]( \frac{0.5}{1} )( \frac{10}{10} )= \frac{5}{10} = \frac{1}{2} [/tex] [tex]( \frac{1.5}{1} )( \frac{10}{10} )= \frac{15}{10} = \frac{3}{2} [/tex]
Since we know that each one of those points is a zero of the function ([tex]x=0[/tex]), we can replace each value into our [tex]x=0[/tex], and work backwards to get our binomials: [tex]x=- \frac{5}{2} [/tex] [tex]2x=-5[/tex] [tex]2x+5=0[/tex] So, our first binomial is [tex](2x+5)[/tex]
[tex]x= \frac{1}{2} [/tex] [tex]2x=1[/tex] [tex]2x-1=0[/tex] So, our second binomial is [tex](2x-1)[/tex]
[tex]x= \frac{3}{2} [/tex] [tex]2x=3[/tex] [tex]2x-3=0[/tex] So, our next binomial is [tex](2x-3)[/tex]
Now lets put all together to find the equation of our function: [tex]y=(2x+5)(2x-1)(2x-3)[/tex] And of course, we can rewrite it in a different order: [tex]y=(2x-1)(2x-3)(2x+5)[/tex]
We can conclude that the equation that most likely represents the graph of the function is [tex]y=(2x-1)(2x-3)(2x+5)[/tex]