GED Mathematical Reasoning TestAlgebraic Problem Solving with Expressions and EquationsMedium
A technician is calibrating a sensor. The sensor's reading (R) in volts is modeled by the equation R = 3x² - 12x + 15, where 'x' is the input signal strength. What is the minimum possible reading of the sensor?
- A30 volts
- B15 volts
- C3 volts
- D12 volts
Show answer & explanationAnswer & explanation
Correct answer: C. 3 volts
The function is a quadratic equation with a positive leading coefficient (a=3), so the parabola opens upwards, and the vertex represents the minimum value. The x-coordinate of the vertex is -b/(2a) = -(-12)/(2*3) = 12/6 = 2. Substitute x=2 into the function: R(2) = 3(2)² - 12(2) + 15 = 3(4) - 24 + 15 = 12 - 24 + 15 = 3 volts.
Why the other options are wrong
- A. Incorrect calculation.
- B. This is the y-intercept, not the minimum reading.
- D. This is the 'b' coefficient (absolute value), not the minimum reading.
Quadratic Function Vertex (Minimum/Maximum)
The vertex of a parabola represents the maximum or minimum value of a quadratic function. For a function in the form ax² + bx + c, the x-coordinate of the vertex is -b/(2a).
- If 'a' > 0, the parabola opens upwards, and the vertex is a minimum.
- If 'a' < 0, the parabola opens downwards, and the vertex is a maximum.
- Substitute the x-coordinate of the vertex back into the function to find the maximum/minimum value.
Memory trick: Vertex finds the highest or lowest peak!