In the xy𝑥𝑦-plane, a parabola has vertex (−13,−23)(-13,-23) and does not intersect the x𝑥-axis. If the equation of the parabola is written in the form y=ax2+bx+c𝑦=𝑎𝑥2+𝑏𝑥+𝑐, where a𝑎, b𝑏, and c𝑐 are constants, which of the following could be the value of a+b+c𝑎+𝑏+𝑐?−25-25eliminate−23-23eliminate00eliminateNo solution
Question
In the xy𝑥𝑦-plane, a parabola has vertex (−13,−23)(-13,-23) and does not intersect the x𝑥-axis. If the equation of the parabola is written in the form y=ax2+bx+c𝑦=𝑎𝑥2+𝑏𝑥+𝑐, where a𝑎, b𝑏, and c𝑐 are constants, which of the following could be the value of a+b+c𝑎+𝑏+𝑐?−25-25eliminate−23-23eliminate00eliminateNo solution
Solution
The vertex form of a parabola is given by y = a(x-h)² + k, where (h,k) is the vertex of the parabola. Given that the vertex is (-13,-23), the equation of the parabola can be written as y = a(x+13)² - 23.
Since the parabola does not intersect the x-axis, it means that the parabola opens upwards or downwards but not across. This implies that the coefficient 'a' is either positive or negative but not zero.
Expanding the equation y = a(x+13)² - 23 gives y = ax² + 26ax + 169a - 23. Comparing this with the standard form y = ax² + bx + c, we get a = a, b = 26a, and c = 169a - 23.
The sum a + b + c then becomes a + 26a + 169a - 23 = 196a - 23. Since 'a' is not zero, the sum a + b + c cannot be zero. Therefore, the possible value of a + b + c could be -25 or -23.
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