This is a quadratic and there are a few tricks that may be used to find salient points for sketching them.
Given: y=-(x-2)(x+5)y=−(x−2)(x+5)
Multiply the brackets giving:
y = -x^2-3x+10y=−x2−3x+10....... (1)
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First off; we have a negative x^2x2. This results in an inverted horse shoe type plot. That is of shape nn∩ instead of U.
Using standard form of y=ax^2+bx+cy=ax2+bx+c
To do the next bit you would need to change this standard form into y=a(x^2 +b/a x + c/a)y=a(x2+bax+ca). It is the bit inside the brackets we are looking at. In your case a=1a=1 so we do not need to change anything.
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color(blue)("The minima for "x "occurs at " -1/2 times b/a")The minima for xoccurs at −12×ba
color(blue)("In your case")In your case
color(blue)(a=1)a=1
color(blue)(b=-3)b=−3
so color(red)(x_("minimum") = (-1/2) times (-3) = + 3/2)xminimum=(−12)×(−3)=+32
Substitute color(red)(x_("minimum"))xminimum in equation (1) giving
color(red)(y= -(3/2)^2-3(3/2)+10 )y=−(32)2−3(32)+10
color(green)("You have now found the values for " (x,y)_("minimum"))You have now found the values for (x,y)minimum
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color(blue)(" To find y-intercept substitute "x=0" in equation (1)") To find y-intercept substitute x=0 in equation (1)
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color(blue)(" To find x-intercepts substitute "y=0" in equation (1)") To find x-intercepts substitute y=0 in equation (1)
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