The graph and equation of a function is given.
It is required to find its y-intercept and zeros using the graph, and then finding these values algebraically.
Using the graph:
Recall that the y-intercept is the point where the graph of the function intersects the y-axis.
From the graph, notice that the graph intersects the y-axis at y=0.
Hence, the y-intercept is y=0.
Recall also that the x-intercepts or zeros is the point where the graph of the function intersects the x-axis.
From the graph, the graph intersects the x-axis at x= -1,0, and 1.5.
Hence, the zeros are x= -1,0, and 1.5.
Find these values algebraically:
To find the y-intercept algebraically, substitute x=0 into the function:
[tex]\begin{gathered} f(x)=2x^3-x^2-3x \\ \text{ Substitute }x=0\text{ into the equation:} \\ \Rightarrow f(0)=2(0)^3-0^2-3(0) \\ \Rightarrow f(0)=0 \end{gathered}[/tex]
Hence, the y-intercept is y=0.
To find the zeros, substitute f(x)=0 into the function and solve for x:
[tex]\begin{gathered} 2x^3-x^2-3x=0 \\ \text{ Factor the left-hand side:} \\ \Rightarrow x(2x^2-x-3)=0 \\ \Rightarrow x(2x^2-3x+2x-3)=0 \\ \Rightarrow x[x(2x-3)+1(2x-3)]=0 \\ \Rightarrow x(x+1)(2x-3)=0 \end{gathered}[/tex]
Equate each factor to 0
