Answer:
[tex](h, k) = (-2, 3)[/tex]
[tex]a = -1[/tex]
[tex]\textsf{Equation:}\quad y = -(x + 2)^2 + 3[/tex]
Average rate of change between x = -1 and x = 0 is -3.
Average rate of change between x = -4 and x = -2 is 2.
Step-by-step explanation:
The given graph shows a downward-opening parabola, which means that the equation of the graph is quadratic.
The vertex of a parabola is the turning point of the graph, which is the maximum point for a downward-opening parabola, and the minimum point for an upward-opening parabola. Therefore, in this case, the vertex is (-2, 3).
The vertex form of a quadratic equation is:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Vertex form of a quadratic equation}}\\\\y=a(x-h)^2+k\\\\\textsf{where:}\\\phantom{ww}\bullet\;(h,k)\;\sf is\;the\;vertex.\\\phantom{ww}\bullet\;a\;\sf is\;the\;leading\;coefficient.\\\end{array}}[/tex]
As the vertex of the parabola is (-2, 3), then:
[tex]\Large\boxed{\boxed{(h,k)=(-2, 3)}}[/tex]
To find the equation of the graphed parabola, first substitute the vertex (1, 1) into the formula:
[tex]y=a(x-(-2))^2+3\\\\\\y=a(x+2)^2+3[/tex]
Now, substitute a point on the curve into the equation and solve for a. Let's use the y-intercept (0, -1):
[tex]-1=a(0+2)^2+3 \\\\\\ -1=a(2)^2+3\\\\\\-1=4a+3\\\\\\4a=-4\\\\\\a=-1[/tex]
So, the leading coefficient is:
[tex]\Large\boxed{\boxed{a=-1}}[/tex]
Substitute the vertex and leading coefficient into the formula to create the equation of the graphed function in vertex form:
[tex]\Large\boxed{\boxed{y=-(x+2)^2+3}}[/tex]
To find the average rate of change over an interval [a, b], we subtract the y-values at these two points, and then divide this difference by the change in x:
[tex]\boxed{\begin{array}{c}\underline{\textsf{Average rate of change of function $f(x)$}}\\\\$\dfrac{f(b)-f(a)}{b-a}$\\\\\textsf{over the interval $[a,b]$}\end{array}}[/tex]
In the first case, the interval is [-1, 0], so:
- a = -1
- b = 0
- f(a) = f(-1)
- f(b) = f(0)
From observation of the graph:
[tex]\textsf{Average rate of change}=\dfrac{f(0)-f(-1)}{0-(-1)}=\dfrac{-1-2}{0+1}=\dfrac{-3}{1}=-3[/tex]
Therefore, the average rate of change between x = -1 and x = 0 is:
[tex]\Large\boxed{\boxed{\textsf{Average rate of change}=-3}}[/tex]
In the second case, the interval is [-4, -2], so:
- a = -4
- b = -2
- f(a) = f(-4)
- f(b) = f(-2)
From observation of the graph:
[tex]\textsf{Average rate of change}=\dfrac{f(-2)-f(-4)}{-2-(-4)}=\dfrac{3-(-1)}{-2+4}=\dfrac{3+1}{2}=\dfrac{4}{2}=2[/tex]
Therefore, the average rate of change between x = -4 and x = -2 is:
[tex]\Large\boxed{\boxed{\textsf{Average rate of change}=2}}[/tex]
