To find the value of $k$ and the coordinates of point $P$ where the graph of $y = kx - k$ touches the graph of $y = 3x^2 - 4x + 1$, we need to follow these steps:

Step 1: Set the equations equal

Since the graphs touch at point $P$, the $y$-values for both equations will be equal at this point. Therefore, we can set the equations equal to each other:

$kx - k = 3x^2 - 4x + 1$

Step 2: Rearrange the equation

Rearrange the equation to bring all terms to one side:

$3x^2 - (4 + k)x + (1 + k) = 0$

This is a quadratic equation in $x$.

Step 3: Condition for touching

For the graphs to touch, the quadratic equation must have exactly one solution (since the curves are tangent). This happens when the discriminant of the quadratic equation is zero. The discriminant $\Delta$ of a quadratic equation $ax^2 + bx + c = 0$ is given by:

$\Delta = b^2 - 4ac$

Here, $a = 3$, $b = -(4 + k)$, and $c = 1 + k$. Substituting these into the discriminant formula:

$\Delta = (-(4 + k))^2 - 4(3)(1 + k)$

Simplifying:

$\Delta = (4 + k)^2 - 12(1 + k)$ $\Delta = (16 + 8k + k^2) - 12 - 12k$ $\Delta = k^2 - 4k + 4$

Step 4: Set the discriminant to zero

For the curves to touch, the discriminant must be zero:

$k^2 - 4k + 4 = 0$

This is a perfect square trinomial:

$(k - 2)^2 = 0$

So, $k = 2$.

Step 5: Find the point $P$

Now that we know $k = 2$, substitute this value into the equation for $y = 2x - 2$ and solve for $x$ and $y$.

Substitute $k = 2$ into the equation $2x - 2 = 3x^2 - 4x + 1$:

$2x - 2 = 3x^2 - 4x + 1$

Rearranging:

$3x^2 - 6x + 3 = 0$

Dividing through by 3:

$x^2 - 2x + 1 = 0$

This is another perfect square trinomial:

$(x - 1)^2 = 0$

So, $x = 1$.

Substitute $x = 1$ into either equation to find $y$. Using $y = 2x - 2$:

$y = 2(1) - 2 = 0$

Thus, the point $P$ is $(1, 0)$.

Conclusion:

• The value of $k$ is $2$.
• The point $P$ where the graphs touch is $(1, 0)$.

Do you need further details or explanations?

Here are 5 related questions:

1. How do you find the intersection point of two graphs?
2. What is the significance of the discriminant in a quadratic equation?
3. How do you find the tangent to a curve at a given point?
4. How would the solution change if the curves intersected at two points?
5. What are other methods to verify if two curves are tangent?

Tip: When solving quadratic equations, always check the discriminant to understand the nature of the roots.