-s2.9 parabola....vertex, open direction, increasing, decreasing, zeros

In summary, the function $g(t)=3t^2-5t+1$ can be converted to vertex form $g(t)=3\left(t-\frac{5}{6}\right)^2-\frac{13}{12}$, where the parabola opens up and the vertex is located at $\left(\frac{5}{6},-\frac{13}{12}\right)$. The interval of increasing is $\left[\frac{5}{6},\infty\right)$ and the interval of decreasing is $\left(-\infty,\frac{5}{6}\right]$. The parabola is commonly used in speed problems and its vertex is at $\left(\frac{5
  • #1
karush
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What are intervals of which the function $g(t)=3t^2-5t+1$
a. find the vertex convert to vertex form $g(t)=a(t-h)^2-k$
\item $g(t)=3\left(t-\dfrac{5}{3}t+\left(\dfrac{5}{6}\right)^2\right)
+1-\dfrac{25}{12}
=3\left(t-\dfrac{5}{6}\right)^2-\dfrac{13}{12}$
b. the parabola opens up and the vertex is $\left(\dfrac{5}{6},-\dfrac{13}{12}\right)$
c. is decreasing or increasing
interval of increasing is $\left[\dfrac{5}{6}<x \right]$
and the interval of decreasing is $\left[x<\dfrac{5}{6}\right]
41.png


ok, hopefully this is correct
typos maybe

the parabola seens to be used a lot in speed problems
 
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  • #2
$g(t)=3t^2−5t+1= 3(t^2- \frac{5}{2}t)+ 1= 3(x^3- \frac{5}{2}x+ \frac{25}{16}- \frac{25}{16})+ 1$​
$g(t)= 3(x- \frac{5}{4})^2- \frac{75}{16}+ 1=$$ 3(x- \frac{5}{4})^2- \frac{59}{4}$.​
The vertex is at $\left(\frac{5}{4}, -\frac{59}{4}\right)$​
 
Last edited:
  • #3
ok just need to be more careful
 

Related to -s2.9 parabola....vertex, open direction, increasing, decreasing, zeros

1. What is the vertex of a parabola?

The vertex of a parabola is the point where the parabola changes direction from increasing to decreasing (or vice versa). It is also the highest or lowest point on the parabola, depending on whether the parabola opens upwards or downwards.

2. How can you determine the open direction of a parabola?

The open direction of a parabola can be determined by looking at the coefficient of the squared term in the equation. If the coefficient is positive, the parabola opens upwards, and if it is negative, the parabola opens downwards.

3. What does it mean for a parabola to be increasing?

A parabola is increasing when its graph moves from left to right in an upward direction. This means that the y-values are getting larger as the x-values increase.

4. How do you know if a parabola is decreasing?

A parabola is decreasing when its graph moves from left to right in a downward direction. This means that the y-values are getting smaller as the x-values increase.

5. What are zeros of a parabola?

Zeros of a parabola are the x-values where the parabola intersects the x-axis. These are also known as the roots or solutions of the quadratic equation that represents the parabola.

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