Answer:
The axis of symmetry is at [tex]x=-1[/tex]
The graph has an x-intercept at [tex](1,0)[/tex]
The graph has a vertex at [tex](-1,4)[/tex]
Step-by-step explanation:
we have
[tex]y=-x^{2}-2x+3[/tex]
Statements
case 1) The graph has root at [tex]3[/tex] and [tex]1[/tex]
The statement is False
Because, the roots of the quadratic equation are the values of x when the value of y is equal to zero (x-intercepts)
Observing the graph, the roots are at [tex]-3[/tex] and [tex]1[/tex]
case 2) The axis of symmetry is at [tex]x=-1[/tex]
The statement is True
Observing the graph, the vertex is the point [tex](-1,4)[/tex]
The axis of symmetry in a vertical parabola is equal to the x-coordinate of the vertex
so
the equation of the axis of symmetry is [tex]x=-1[/tex]
case 3) The graph has an x-intercept at [tex](1,0)[/tex]
The statement is True
see procedure case 1)
case 4) The graph has an y-intercept at [tex](-3,0)[/tex]
The statement is False
Because, the y-intercept is the value of y when the value of x is equal to zero
Observing the graph, the y-intercept is the point [tex](0,3)[/tex]
case 5) The graph has a relative minimum at [tex](-1,4)[/tex]
The statement is False
Because, is a vertical parabola open downward, therefore the vertex is a maximum
The point [tex](-1,4)[/tex] represent the vertex of the parabola, so is a maximum
case 6) The graph has a vertex at [tex](-1,4)[/tex]
The statement is True
see the procedure case 5)
see the attached figure to better understand the problem
