The graph of the function f(x) = px? - x + 3 does not touch
or cross the x-axis. What are the possible values for p?

The function \( f(x) = px^2 - x + 3 \) is a quadratic equation, and we are told that its graph does not touch or cross the x-axis. This means that the quadratic has no real solutions, and thus its discriminant must be **negative**.

For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant is given by:

\[
\Delta = b^2 - 4ac
\]

In this case:
- \( a = p \),
- \( b = -1 \),
- \( c = 3 \).

Substitute these values into the discriminant formula:

\[
\Delta = (-1)^2 - 4(p)(3) = 1 - 12p
\]

For the quadratic to have no real solutions, we need \( \Delta < 0 \):

\[
1 - 12p < 0
\]

Solve this inequality:

\[
1 < 12p
\]

\[
p > \frac{1}{12}
\]

Thus, the function \( f(x) = px^2 - x + 3 \) does not touch or cross the x-axis when \( p > \frac{1}{12} \).

### Would you like more details or have any questions?

Here are 5 related questions you might want to explore:
1. What happens to the function when \( p = \frac{1}{12} \)?
2. How do we find the vertex of a quadratic function?
3. What is the significance of the discriminant in quadratic equations?
4. How does changing the value of \( p \) affect the shape of the parabola?
5. What are the x-intercepts of a quadratic equation?

**Tip:** The discriminant is a useful tool for quickly determining the nature of the roots of any quadratic equation.

The graph of the function f(x) = px^2 - x + 3 does not touch or cross the x-axis. What are the possible values for p?

Discover how to find the possible values of p for the quadratic equation f(x) = px^2 - x + 3, where the graph does not touch or cross the x-axis. This problem involves the discriminant and inequalities, perfect for students in Grades 9-12.

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