Find the value of $k$ & Find the roots of the equation

Question

The graph of quadratic function drawn on the interval $-1\leq x\leq 5$.

i.If the quadratic function is , $y=(x-2)^2-k$, find the value of $k$.

ii.for this value of $k$, find the roots of the equation $(x-2)^2-k=0$

Any Ideas on how to begin?

Answer 1

We know that the roots of this graph are going to be where it intersects the x-axis. It does this at x = 0 and x = 4. Also, we know that this is a parabola, so it has two roots.

Hence, $f(x) = x(x-4) = x^2 - 4x$

Now you want to find a value of $k$ satisfying $(x-2)^2-k = x^2 - 4x$

$(x-2)^2 - k = x^2-4x+4 -k = x^2-4x$ implies $4 = k$

Answer 2

1) V, the vertex, is at $(2, -4)$. That is, when $x = 2, y = -4$.

When $x = 2, y = (2 – 2)^2 – k = - k$. Therefore, k = 4.

2) Solving $(x – 2)^2 – k = 0$ is equivalent to solving

$y = (x-2)^2 – k $; AND $y = 0$.

The roots are given by the points of intersection of these two graphs. One has been clearly shown but the graph of $y = 0$ requires an add-on.

Answer 3

i.If the quadratic function is , $y=(x-2)^2-k$, find the value of k.

Finding $k$ is easy. You have the function $y=f(x)$ and its graph. These two should be equivalent. That is, if the point $(x,y)$ is on the curve, it should also satisfy the equation $y=f(x)$, and vice versa. Now, to determine $k$, you can pick any point from the curve and form the mentioned equation for it.

For example, pick the point $(1,-3)$. Thus $$-3=(1-2)^2-k$$ which results in $$k=4$$You can pick any other point of the curve and the result would be the same (check it!).

ii.for this value of k, find the roots of the equation $ (x-2)^2-k=0$

Now, with $k=4$, the equation is $(x-2)^2-4=0$. You can solve it in different ways, including the general method (i.e., the $\Delta$ rule). By expanding the binomial square you get $$\begin{aligned}(x-2)^2-4 & = 0 \\ x^2-4x+4-4 & = 0 \\ x^2-4x & = 0\end{aligned}$$ And by factoring $x$ $$x(x-4)=0$$ This means either $x$ or $x-4$ should be zero. So there are two solutions: $x=0$ and $x=4$. Now take a look at the graph. What is happening for the graph at $x=0$ or $x=4$? In other words, what are $y$ values at $x=0$ and $x=4$?

(The whole point of the question was to show you the relation between a quadratic function formula and its graph.)