Question4. a) Explain how you would solve this problem: For what value of does the function have only one zero? (1 mark) b) Use your strategy to find the value of . (3 marks)
Studdy Solution
STEP 1
1. The function is a quadratic function.
2. We need to find the value of for which this function has exactly one zero.
3. A quadratic function has one zero when its graph touches the x-axis at exactly one point (i.e., when it has a tangent point with the x-axis).
STEP 2
1. Explain the strategy to solve the problem
2. Apply the strategy to find the value of
3. Verify the solution
STEP 3
To solve this problem, we need to consider the discriminant of the quadratic function. The discriminant determines the nature of the roots of a quadratic equation. For a quadratic function to have exactly one zero, its discriminant must be equal to zero. This occurs when the parabola just touches the x-axis at one point.
The discriminant of a quadratic equation is given by .
Our strategy will be to:
a) Identify the coefficients , , and in our function.
b) Set up the discriminant equation and set it equal to zero.
c) Solve the resulting equation for .
STEP 4
Let's apply our strategy:
a) In , we can identify:
b) Now, let's set up the discriminant equation and set it to zero:
c) Solve for :
STEP 5
To verify our solution, we can substitute back into the original function:
Now, if we were to solve this equation, we should get exactly one solution. We can confirm this by looking at the discriminant again:
Since the discriminant is zero, this confirms that the function has exactly one zero when .
Therefore, the function has only one zero when:
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