Using the equation [tex]y = ax^2 + bx + c[/tex] to represent a parabola on a graph, which statement is true?
1. If [tex]b[/tex] is negative, the parabola opens downward.
2. If [tex]a[/tex] is negative, the parabola opens upward.
3. If [tex]a[/tex] is positive, the parabola opens upward.
4. If [tex]c[/tex] is negative, the parabola opens downward.
Answer (3)
Tha answer is 3) if a is positive, the parabola opens upward. You can see that when you imagine what happens when x is large. In that case bx + c is small compared to ax^2, and the value of the function is dominated by the terms ax^2. x^2 is positive always, so the sign of ax^2 is given by a. Then if a is positive the extreme values of the function (x positively large and x negatively large) are positive, which is the case when the parabola opens upward.
The true statement is that if a is positive, the parabola opens upward. The coefficient a determines the direction of the parabola's opening.
The standard form of a quadratic equation representing a parabola is y=ax^2 +bx+c.
The correct statements regarding the behavior of the parabola based on the coefficients are as follows:
If a is negative, the parabola opens downward. The coefficient a determines the direction of the opening. If a is negative, the parabola opens downward, and if a is positive, the parabola opens upward.
If a is positive, the parabola opens upward. This statement is correct as stated above.
If b is negative, it affects the axis of symmetry but doesn't determine the direction of the opening. Therefore, this statement is not accurate in determining the direction of the parabola's opening.
If c is negative, it shifts the parabola vertically but does not determine the direction of the opening.
Therefore, the correct statements regarding the behavior of the parabola based on the coefficients are:
If a is negative, the parabola opens downward.
If a is positive, the parabola opens upward.
The correct statement is that if a is positive, the parabola opens upward. The coefficient a determines the direction of the parabola's opening, while b and c affect position and translation. Hence, option 3 is the true statement.
;
