Question

I’m lost on trying to help my son with this problem. Any helps appreciated

Answer 1

Answer:

a) the line of symmetry for f(x) and g(x) is the same (x = -3)

b) The y-intercept of f is greater than the y-intercept of g

c) the average rate of change for f is less than the rate of change of g

Step-by-step explanation:

Given:

A table describing functon f(x) and a parabola describing function g(x)

To find:

we need to complete the statements by filling in the blanks

a) The first is to determine the line of symmetry of f and g

The line of symmetry is the value of x which divides the parabola into equal halves (mirror images)

It is also the x coordinate of the vertex of the parabola. Since we are comparing f and g, it means plotting the points on the table will give a parabola.

Graph of f(x) showing the line of symmetry:

Graph of function g(x):

From the digrams above, the line of symmetry for f(x) and g(x) is the same (x = -3)

b) the y-intercept is the value of y when x = 0

On a graph, it is the value of y when the line crosses the y axis

On the graph of f(x), the line crosses the y axis at y = -2

Hence, the y-intercept is -2

On the table for f(x), x = 0 when f(x) = 8

On graph of f(x), the line crosses the y axis at y = 8

Hence, the y-intercept is 8

8 > -2

The y-intercept of f is greater than the y-intercept of g

c) The rate of change of a function is given as:

`\begin{gathered} rate\text{ of change = }\frac{change\text{ in y axis}}{change\text{ in x axis for the interval}}\text{ } \\ \\ rate\text{ of change for f\lparen x\rparen = }\frac{f(b)\text{ - f\lparen a\rparen}}{b\text{ - a}} \\ for\text{ interval a < x < b, that is be is reater than a} \end{gathered}`
`\begin{gathered} Interval\text{ = \lbrack-6, -3\rbrack} \\ when\text{ x = -6, f\lparen x\rparen = 8} \\ when\text{ x = -3, f\lparen x\rparen = -10} \\ rate\text{ of change =}\frac{-10\text{ - 8}}{-3-(-6)} \\ rate\text{ of change = }(-18)/(-3+6)\text{ = }(-18)/(3) \\ rate\text{ of change = -6} \end{gathered}`

`\begin{gathered} rate\text{ of change for g\lparen x\rparen = }\frac{g(b)\text{ - g\lparen a\rparen}}{b\text{ - a}} \\ for\text{ interval: \lbrack-6, -3\rbrack} \\ rate\text{ of change for g\lparen x\rparen= }\frac{g(-3)\text{ - g\lparen-6\rparen}}{-3\text{ - \lparen-6\rparen}} \\ \\ when\text{ x = -6, g\lparen x\rparen = -2 } \\ when\text{ x = -3, g\lparen x\rparen = 7} \\ rate\text{ of change for g\lparen x\rparen= }\frac{7\text{ - \lparen-2\rparen}}{-3-(-6)}=\text{ }(7+2)/(-3+6) \\ rate\text{ of change for g\lparen x\rparen = }(9)/(3) \\ rate\text{ of change for g\lparen x\rparen= 3} \end{gathered}`

rate of change for f against g: -6 < 3

Over the interval [-6, -3], the average rate of change for f is less than the rate of change of g