SOLUTION: Find the vertex and intercepts for each parabola. Sketch the graph. G(x) = x² + x – 6

Algebra ->  Graphs -> SOLUTION: Find the vertex and intercepts for each parabola. Sketch the graph. G(x) = x² + x – 6       Log On


   

Question 150542: Find the vertex and intercepts for each parabola. Sketch the graph.
G(x) = x² + x – 6

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Vertex:



In order to find the vertex, we first need to find the x-coordinate of the vertex.


To find the x-coordinate of the vertex, use this formula: x=%28-b%29%2F%282a%29.


x=%28-b%29%2F%282a%29 Start with the given formula.


From y=x%5E2%2Bx-6, we can see that a=1, b=1, and c=-6.


x=%28-%281%29%29%2F%282%281%29%29 Plug in a=1 and b=1.


x=%28-1%29%2F%282%29 Multiply 2 and 1 to get 2.


So the x-coordinate of the vertex is x=-1%2F2. Note: this means that the axis of symmetry is also x=-1%2F2.


Now that we know the x-coordinate of the vertex, we can use it to find the y-coordinate of the vertex.


y=x%5E2%2Bx-6 Start with the given equation.


y=%28-1%2F2%29%5E2-1%2F2-6 Plug in x=-1%2F2.


y=1%2F4-1%2F2-6 Square -1%2F2 to get 1%2F4.


y=-25%2F4 Combine the fractions.


So the y-coordinate of the vertex is y=-25%2F4.


So the vertex is which in decimal form is (-0.5, -6.25).






Intercepts:

X-intercept(s):

y=x%5E2%2Bx-6 Start with the given equation.


0=x%5E2%2Bx-6 Plug in y=0


Let's use the quadratic formula to solve for x


x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


x+=+%28-%281%29+%2B-+sqrt%28+%281%29%5E2-4%281%29%28-6%29+%29%29%2F%282%281%29%29 Plug in a=1, b=1, and c=-6


x+=+%28-1+%2B-+sqrt%28+1-4%281%29%28-6%29+%29%29%2F%282%281%29%29 Square 1 to get 1.


x+=+%28-1+%2B-+sqrt%28+1--24+%29%29%2F%282%281%29%29 Multiply 4%281%29%28-6%29 to get -24


x+=+%28-1+%2B-+sqrt%28+1%2B24+%29%29%2F%282%281%29%29 Rewrite sqrt%281--24%29 as sqrt%281%2B24%29


x+=+%28-1+%2B-+sqrt%28+25+%29%29%2F%282%281%29%29 Add 1 to 24 to get 25


x+=+%28-1+%2B-+sqrt%28+25+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


x+=+%28-1+%2B-+5%29%2F%282%29 Take the square root of 25 to get 5.


x+=+%28-1+%2B+5%29%2F%282%29 or x+=+%28-1+-+5%29%2F%282%29 Break up the expression.


x+=+%284%29%2F%282%29 or x+=++%28-6%29%2F%282%29 Combine like terms.


x+=+2 or x+=+-3 Simplify.


So the solutions are x+=+2 or x+=+-3

This means that the x-intercepts are (-3,0) and (2,0)

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Y-intercept:

y=x%5E2%2Bx-6 Start with the given equation.


y=%280%29%5E2%2B0-6 Plug in x=0.


y=1%280%29%2B0-6 Square 0 to get 0.


y=0%2B0-6 Multiply 1 and 0 to get 0.


y=-6 Combine like terms.


So the y-intercept is (0,-6)





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Summary:

So the vertex is which in decimal form is (-0.5, -6.25)

The x-intercepts are (-3,0) and (2,0) and the y-intercept is (0,-6)



Plot these points on the coordinate system






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Graph:


Now draw a parabola through all of the points to graph y=x%5E2%2Bx-6:


Graph of y=x%5E2%2Bx-6