We are tasked with analyzing the quadratic equation $y = kx^2 - 6x + k$ to determine whether there are two possible values for $k$ such that the graph of this equation touches the $x$-axis at only one point.

Step-by-Step Solution:

1. Key Condition for Touching the $x$-axis:

   • A quadratic equation $ax^2 + bx + c = 0$ touches the $x$-axis at only one point if it has exactly one real root. This happens when the discriminant ($\Delta$) of the quadratic equation is zero: $\Delta = b^2 - 4ac = 0.$

2. Identify the Coefficients: From $y = kx^2 - 6x + k$, the coefficients are: $a = k, \quad b = -6, \quad c = k.$

3. Set the Discriminant to Zero: Substitute $a = k$, $b = -6$, and $c = k$ into the discriminant formula $\Delta = b^2 - 4ac$: $\Delta = (-6)^2 - 4(k)(k).$ Simplify: $\Delta = 36 - 4k^2.$ For the graph to touch the $x$-axis at one point, set $\Delta = 0$: $36 - 4k^2 = 0.$

4. Solve for $k$: Rearrange the equation: $4k^2 = 36.$ Divide through by 4: $k^2 = 9.$ Solve for $k$: $k = \pm 3.$

5. Interpretation:

   • There are two possible values for $k$: $k = 3$ and $k = -3$.
   • For each of these values, the graph of the equation will touch the $x$-axis at exactly one point.

Verification:

For $k = 3$ and $k = -3$, substitute back into the equation and confirm:

• When $k = 3$, $y = 3x^2 - 6x + 3$. This simplifies to: $y = 3(x^2 - 2x + 1) = 3(x - 1)^2.$ The graph is a parabola with its vertex at $(1, 0)$, touching the $x$-axis at $x = 1$.

• When $k = -3$, $y = -3x^2 - 6x - 3$. This simplifies to: $y = -3(x^2 + 2x + 1) = -3(x + 1)^2.$ The graph is a parabola with its vertex at $(-1, 0)$, touching the $x$-axis at $x = -1$.

Final Answer:

Yes, there are two possible values for $k$: $k = 3$ and $k = -3$.

Would you like further clarification on this solution?

Related Questions:

1. How do we determine the vertex of a quadratic equation like $y = kx^2 - 6x + k$?
2. What happens to the graph of the quadratic if $\Delta > 0$ or $\Delta < 0$?
3. How does the sign of $k$ affect the direction of the parabola?
4. Can a quadratic equation touch the $x$-axis at more than one point?
5. How would the problem change if $k = 0$?

Tip:

Always use the discriminant $\Delta = b^2 - 4ac$ to analyze the roots of quadratic equations—it helps quickly determine the nature of the solutions!