MHB Max or Min curve on a graph question

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The discussion focuses on solving the quadratic equation x² + 5x - 6, where the roots are found to be x = 6 and x = -1 using the quadratic formula. The graph of the function is sketched, indicating that it crosses the x-axis at -3 and 6, with a minimum point at (-2.5, -12.25). The derivative confirms the minimum value of the curve, as the second derivative is positive. The equation of the tangent line at x = 2 is discussed, with participants clarifying the formula needed and the values to substitute. Overall, the conversation emphasizes the importance of understanding the properties of quadratic functions and their graphical representations.
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a) Find the roots of the equation $$x^{2}+5x-6$$

b) Sketch the graph of the function $$x^{2}+5x-6$$ labeling the points at which the graph crosses the axes and the co-ordinates of the maximum and minimum of the curve

c) Find the equation of the tangent at the point where $$x=2$$ on the curve of $$y=x^{2}+5x-6$$

MY SOLUTION Right so far?

a) Using the quadratic formula, we get $$x=\frac{-5\pm\sqrt{49}}{2}$$

$$\therefore x=6$$ or $$-1$$

b)
$$y=x^{2}+5x-6$$

$$\d{y}{x}$$ = $$2x+5$$

$$x=-\frac{5}{2}$$ (-2.5)

Sub x into equation

$$y=(-\frac{5}{2})^{2}+5(-\frac{5}{2})-6$$

y=$$-\frac{49}{4}$$ (-12.25)

and $$\d{y^{2}}{^{2}x}$$ = 2 which is a minimum value

So with the graph, you would plot it with the parabola going with the x points -3 and 6 and the y points $$-\frac{49}{4}$$

c) No clue!
 
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mathsheadache said:
a) Find the roots of the equation $$x^{2}+5x-6$$
Note that this is not an equation because an equation must have the = sign. You could find the roots of the equation $$x^{2}+5x-6=0$$ or of polynomial $$x^{2}+5x-6$$.

mathsheadache said:
a) Using the quadratic formula, we get $$x=\frac{-5\pm\sqrt{49}}{2}$$

$$\therefore x=6$$ or $$-1$$
I recommend substituting these values back into the original equation and checking if they are indeed roots.

mathsheadache said:
b)
$$y=x^{2}+5x-6$$

$$\d{y}{x}$$ = $$2x+5$$

$$x=-\frac{5}{2}$$ (-2.5)

Sub x into equation

$$y=(-\frac{5}{2})^{2}+5(-\frac{5}{2})-6$$

y=$$-\frac{49}{4}$$ (-12.25)

and $$\d{y^{2}}{^{2}x}$$ = 2 which is a minimum value
I agree. It may make sense to remember that the $x$-coordinate of the vertex of the parabola $y=ax^2+bx+c$ is $-b/(2a)$ and that the type of extremum is determined by the sign of $a$: if $a>0$, then the function has a minimum and if $a<0$, then it has a maximum.

mathsheadache said:
So with the graph, you would plot it with the parabola going with the x points -3 and 6 and the y points $$-\frac{49}{4}$$
This is stated a little vague.

mathsheadache said:
c) Find the equation of the tangent at the point where $$x=2$$ on the curve of $$y=x^{2}+5x-6$$
The equation of the tangent to $y=f(x)$ at point $(x_0,f(x_0))$ is $y-y_0=f'(x_0)(x-x_0)$.
 
The equation of the tangent to $y=f(x)$ at point $(x_0,f(x_0))$ is $y-y_0=f'(x_0)(x-x_0)$.

Is this the formula? What values would you need to sub in?
 
mathsheadache said:
Is this the formula?
Well, it's not an apple! (Smile) You can see it in Wikipedia.

mathsheadache said:
What values would you need to sub in?
I think my post mentions everything that one needs to use this formula. I suggest you re-read it, and if you have further questions, feel free to describe what you don't understand.
 
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