Answer:
1) Vertex point (-1.5,-20.25)
2) axis of symmetry: x=-1.5
3) x-intercepts: (-6,0) and (3,0)
4) y-intercept: (0,18)
5) Graph in attachment.
Step-by-step explanation:
1) The given parabola has equation:
[tex]y = {x}^{2} + 3x - 18[/tex]
We need to complete the square.
[tex]y = {x}^{2} + 3x + {1.5}^{2} - {1.5}^{2} - 18[/tex]
[tex]y = ( {x + 1.5)}^{2} - 20. 25[/tex]
This function is in the vertex form:
[tex]y = a( {x - h)}^{2} + k[/tex]
where (h,k) is the vertex, x=h is the axis of symmetry.
By comparing our equation to the general vertex form:
The vertex point is (-1.5,-20.25)
2) The axis of symmetry divides the parabola into two congruent halves.
Since this is a vertical parabola, the axis of symmetry occuring at x=h is a vertical line.
The axis of symmetry is x=-1.5
3) Y-INTERCEPT.
The y-intercept is the point where the graph cross the y-axis.
At this point, the value of x is zero.
To find the y-intercept, we substitute x=0 in the equation of the parabola and simplify.
when x=0,
[tex]y = {(0)}^{2} + 3(0) - 18 = - 18[/tex]
The y-intercept is (0,-18).
4) X-INTERCEPT
The x-intercepts are the points where the graph touches or intersect the x-axis.
To find the x-intercept, we substitute y=0.
[tex]( {x + 1.5)}^{2} - 20.25 = 0[/tex]
Add 20.25 to both sides;
[tex]( {x + 1.5)}^{2} = 20.25 [/tex]
Take square root.
[tex](x + 1.5)= \pm \sqrt{20.25} [/tex]
[tex]x = - 1.5\pm 4.5 [/tex]
[tex]x = - 1.5 - 4.5 \: \: or \: \: x = - 1.5 + 4.5[/tex]
[tex]x = - 6 \: or \: x = 3[/tex]
The x-intercepts are(-6,0) and (3,0).
4) GRAPH
To graph this function, we can use transformation.
To graph the function,
[tex]y = {(x + 1.5)}^{2} - 20.25[/tex]
We shift the parent quadratic function 1.5 units left and 20.25 units down.
See attachment for graph.
