Which three options are true about the quadratic equation??
y= 4t^2 + 16t + 18
A. The graph has a minimum.
B. The graph of y crosses the horizontal axis at t = 10 and t = 20.
C. The equation 4t^2 + 16 + 18 = 6 has one real solution.
D. The function has a maximum at t = 10
E. The vertex is at (-2, 2)
F. The function could represent the height y (in metres) of a particle moving under gravity, as a function of the time t (in seconds).
G. The graph intercepts the vertical axis at y = 18
H. The slope of the graph of y is always negative.
I think the answers are G, E and A. Is this correct
consider 4t^2 + 16t + 18=6
4t^2 + 16t + 18-6=0
b^2-4ac = 256-16*12 is a positive, so there are two real solutions.
so c is wrong.
To determine which options are true about the quadratic equation y = 4t^2 + 16t + 18, let's carefully evaluate each statement:
A. The graph has a minimum.
To understand this, we need to consider the coefficient of the t^2 term, which is positive (4). A positive coefficient for t^2 indicates that the parabola opens upwards, suggesting it has a minimum point. Therefore, option A is true.
B. The graph of y crosses the horizontal axis at t = 10 and t = 20.
To find the x-intercepts (when y = 0), we set the equation equal to 0 and solve for t:
4t^2 + 16t + 18 = 0
However, this quadratic equation does not factor nicely. But we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac))/(2a)
For our equation, a = 4, b = 16, and c = 18. Plugging these values into the quadratic formula will give us the values of t for which y crosses the horizontal axis. Upon calculation, we find that the roots are complex numbers, which means the parabola does not cross the horizontal axis at any point. So, option B is false.
C. The equation 4t^2 + 16 + 18 = 6 has one real solution.
We have already addressed this equation above. It does not have one real solution; rather, it has two complex solutions. Thus, option C is false.
D. The function has a maximum at t = 10.
To find the maximum or minimum value of a quadratic function, we can use the vertex formula:
t = -b/(2a)
From the given equation, we can determine that a = 4 and b = 16. Plugging these values into the vertex formula, we find that the t-value of the vertex is t = -16/(2*4) = -2. However, this does not match t = 10 from option D, so it is false.
E. The vertex is at (-2, 2).
As determined in the previous analysis, the t-value of the vertex is -2. To find the y-value (or the value of y at the vertex), we substitute t = -2 into the equation:
y = 4t^2 + 16t + 18
y = 4(-2)^2 + 16(-2) + 18
This simplifies to y = 2, which matches the statement given in option E. Thus, it is true.
F. The function could represent the height y (in meters) of a particle moving under gravity as a function of time t (in seconds).
The function y = 4t^2 + 16t + 18 does not explicitly indicate a physical relationship with height and time. Without additional context or information, we cannot definitively say whether option F is true or false. Hence, it remains unclear.
G. The graph intercepts the vertical axis at y = 18.
When t = 0, the equation evaluates as:
y = 4(0)^2 + 16(0) + 18
This simplifies to y = 18, confirming that the graph intercepts the vertical axis at y = 18. Therefore, option G is true.
H. The slope of the graph of y is always negative.
The slope of a quadratic graph can vary. It is not necessarily always negative. Thus, option H is false.
Based on this analysis, the options that are true about the quadratic equation y = 4t^2 + 16t + 18 are A, E, and G.