Answer:
1. (7,-9)
2. (4,0) and (10,0)
3. (3,0) and (-3,0)
4. (0,-9)
5. (-1,-32)
Step-by-step explanation:
To find the vertex use the formula
[tex]\frac{-b}{2a}[/tex]
First make the equation into a quadratic equation.
[tex](x-4)(x-10)\\x^{2} -10x-4x+40\\x^{2} -14x+40[/tex]
Now it is in the form
[tex]ax^{2} +bx+c=0[/tex]
Now we can substitute the values
[tex]\frac{-(-14)}{2(1)} \\\frac{14}{2} \\7[/tex]
Now we have 7 as our x value for the vertex we can substitute for y,
[tex]x^{2} -14x+40=y\\(7)^{2} -14(7)+40=y\\49-98+4=y\\y=-9[/tex]
So the vertex for 1 is (7,-9)
To find the intercepts for 2 & 3, they basically already give it to you. You just need to find a value for x for x-4 that will equal it to 0 and another for x for x-10 that will equal it to 0.
[tex]x-4=0\\x=4\\\\x-10=0\\x=10[/tex]
4 is slightly different but start the same,
[tex](x-3)(x+3)\\x^{2}+3x-3x-9\\x^{2}-9[/tex]
Here, c (9) just shows the graph move down 9, so the y intercept = 9.
5 is the same process as 1,
[tex]2(x-3)(x+5)\\2(x^{2}+5x-3x-15)\\2(x^{2}+2x-15)\\2x^{2}+4x-30[/tex]
Vertex,
[tex]\frac{-(4)}{2(2)} \\\frac{-4}{4} \\-1[/tex]
Substitute,
[tex]2x^{2}+4x-30\\2(-1)^{2}+4(-1)-30\\2-4-30\\-32[/tex]
So the vertex is (-1,-32)
