The tangent to f(x)=ax^{2}+bx-7 at the point (1,10) has a gradient of 1

We are given the function \( f(x) = ax^2 + bx - 7 \), and we know that the tangent to this function at the point \( (1,10) \) has a gradient of 1.

To find the values of \( a \) and \( b \), we can use the following conditions:

1. **Condition 1 (Point on the curve):**
   Since the point \( (1, 10) \) lies on the curve, we substitute \( x = 1 \) and \( f(x) = 10 \) into the equation for \( f(x) \):
   \[
   f(1) = a(1)^2 + b(1) - 7 = 10
   \]
   Simplifying:
   \[
   a + b - 7 = 10 \quad \text{or} \quad a + b = 17 \quad \text{(Equation 1)}
   \]

2. **Condition 2 (Gradient of the tangent):**
   The gradient of the tangent to the curve is given by the derivative of \( f(x) \). The derivative is:
   \[
   f'(x) = 2ax + b
   \]
   At \( x = 1 \), the gradient is 1, so we substitute \( x = 1 \) into \( f'(x) \):
   \[
   f'(1) = 2a(1) + b = 1
   \]
   Simplifying:
   \[
   2a + b = 1 \quad \text{(Equation 2)}
   \]

### Solving the system of equations:
We now solve the system of two equations:
1. \( a + b = 17 \)
2. \( 2a + b = 1 \)

Subtract Equation 1 from Equation 2 to eliminate \( b \):
\[
(2a + b) - (a + b) = 1 - 17
\]
\[
a = -16
\]

Now substitute \( a = -16 \) into Equation 1:
\[
-16 + b = 17
\]
\[
b = 33
\]

Thus, the values of \( a \) and \( b \) are \( a = -16 \) and \( b = 33 \).

### Conclusion:
The function is \( f(x) = -16x^2 + 33x - 7 \).

Would you like a detailed explanation of how derivatives are used to find the gradient of the tangent, or do you have any questions?

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Here are five related questions to explore further:

1. How do you find the tangent to a curve at a given point?
2. What does the derivative of a function represent in geometric terms?
3. How can you determine the equation of the tangent line at a specific point?
4. What is the significance of the gradient of a function at a point?
5. How do changes in the coefficients \( a \) and \( b \) affect the shape of the quadratic function?

**Tip:** When finding the tangent line to a function, always check the derivative to determine the slope at the point in question.

Learn how to find the tangent to the quadratic function f(x) = ax^2 + bx - 7 at the point (1,10) with a gradient of 1. This problem involves key algebraic concepts like quadratic functions and differentiation, suitable for students in Grades 10-12.

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