Question

the graph of a quadratic function displays output values that approach positive infinity. what do you know about the leading coefficient of this function?

the leading coefficient is negative.

the leading coefficient is positive.

Explanation:

Analyze the behavior of the quadratic function

Using the Quadratic Functions knowledge point, we represent a general quadratic function in standard form:

$$f(x) = ax^2 + bx + c$$

where \(a
eq 0\) is the leading coefficient. The problem states that the output values (the \(y\)-values) approach positive infinity:

$$y \to \infty$$

Determine the parabola direction

Using the Parabola Direction knowledge point, we analyze how the direction of opening relates to the output values. Since the outputs approach positive infinity, the parabola must open upwards. If it opened downwards, the outputs would approach negative infinity.

Relate direction to the leading coefficient

Using the Leading Coefficient knowledge point, we determine the sign of \(a\). A parabola opens upwards if and only if its leading coefficient is positive:

$$a > 0$$

Therefore, the leading coefficient of this function must be positive.

Answer:

• The leading coefficient is negative.
• The leading coefficient is positive. (Correct answer)