The diagram below shows part of the graph of the quadratic function f(x)=a(x-h)(x-k) with the condition h<k. Point P is the m
inimum point for the graph of the quadratic function.
(a) calculate the values of
(i)h, (ii)k, (iii)a.
(b) determine the symmetry axis equation.
(c) state the coordinates of point P.
(a) calculate the values of
(i)h, (ii)k, (iii)a.
(b) determine the symmetry axis equation.
(c) state the coordinates of point P.
2 answers:
Answer:
see explanation
Step-by-step explanation:
(a)
The zeros are x = 1 and x = 5, thus the factors are (x - 1) and (x - 5)
f(x) = a(x - 1)(x - 5)
To find a substitute (0, 15), the coordinates of the y- intercept into the equation.
15 = a(0 - 1)(0 - 5) = a(- 1)(- 5) = 5a ( divide both sides by 5)
a = 3
Thus h = 1, k = 5, a = 3 and
f(x) = 3(x - 1)(x - 5)
(b)
The axis of symmetry is a vertical line with equation x = c ( c is a constant )
The axis of symmetry passes through the midpoint of the zeros
x = =
= 3
Equation of axis of symmetry is x = 3
(c)
The axis of symmetry passes through P, with x- coordinate x = 3
Substitute x = 3 into the function for corresponding value of y
f(3) = 3(3 - 1)(3 - 5) = 3(2)(- 2) = - 12
Coordinates of P = (3, - 12 )
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Answer:
Domain is all real numbers.
The range is
The function is increasing over .
The function is decreasing over
The function has a positive y-intercept.
-----------------This is a guess if I had interpreted your choices correctly:
Second option: The range is
Third option: The function is increasing over .
Last option: The function has a positive y-intercept.
I can't really read some of your choices. So you can read my above and determine which is false. If you have a question about any of what I said above please let me know.
Note: I guess those 0's are suppose to be infinities? I hopefully your function is .
Step-by-step explanation:
is a polynomial function which mean it has domain of all real numbers. All this sentence is really saying is that there exists a number for any value you input into
.
Now since the is a quadratic then it is a parabola. We know it is a quadratic because it is comparable to ,
.
This means the graph sort of looks like a U or an upside down U.
It is U, when .
It is upside down U, when .
So here we have so
which means the parabola is an upside down U.
Let's look at the range. We know the vertex is either the highest point (if ) or the lowest point (if
).
The vertex here will be the highest point, again since .
The vertex's x-coordinate can be found by evaluating :
.
So the y-coordinate can be found by evaluate for
:
So the highest y-coordinate is 6. The range is therefore .
If you picture the upside down U in your mind and you know the graph is symmetrical about x=-2.
Then you know the parabola is increasing on and decreasing on
.
So let's look at the intevals they have:
So on the function is increasing.
Looking on the function is increasing on (-4,-2) but decreasing on the rest of that given interval.
The function's y-intercept can be found by putting 0 in for :
The y-intercept is positive since 2>0.