Write the Equation of a Polynomial Function Based on Its Graph - Pre-Calculus
Card 0 of 7
Write the quadratic function for the graph:
Write the quadratic function for the graph:
Method 1:
The x-intercepts are 
. These values would be obtained if the original quadratic were factored, or reverse-FOILed and the factors were set equal to zero.
For 
, 
. For 
, 
. These equations determine the resulting factors and the resulting function; 
.
Multiplying the factors and simplifying,
.
Answer: 
.
Method 2:
Use the form 
, where 
is the vertex.
is 
, so 
, 
.
Answer: 
Method 1:
The x-intercepts are
For
Multiplying the factors and simplifying,
Answer:
Method 2:
Use the form
Answer:
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Write the quadratic function for the graph:
Write the quadratic function for the graph:
Because there are no x-intercepts, use the form 
, where vertex 
is 
, so 
, 
, which gives
Because there are no x-intercepts, use the form
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Which could be the equation for this graph?
Which could be the equation for this graph?
This graph has zeros at 3, -2, and -4.5. This means that 
, 
, and 
. That last root is easier to work with if we consider it as 
and simplify it to 
. Also, this is a negative polynomial, because it is decreasing, increasing, decreasing and not the other way around.
Our equation results from multiplying 
, which results in 
.
This graph has zeros at 3, -2, and -4.5. This means that
Our equation results from multiplying
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Write the equation for the polynomial in this graph:
Write the equation for the polynomial in this graph:
The zeros for this polynomial are 
.
This means that the factors are equal to zero when these values are plugged in for x.
multiply both sides by 2
so one factor is 
multiply both sides by 3
so one factor is 
so one factor is 
Multiply these three factors:
The zeros for this polynomial are
This means that the factors are equal to zero when these values are plugged in for x.
Multiply these three factors:
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Write the equation for the polynomial shown in this graph:
Write the equation for the polynomial shown in this graph:
The zeros of this polynomial are 
. This means that the factors equal zero when these values are plugged in.
One factor is 
One factor is 
The third factor is equivalent to 
. Set equal to 0 and multiply by 2:
Multiply these three factors:
The graph is negative since it goes down then up then down, so we have to switch all of the signs:
The zeros of this polynomial are
One factor is
One factor is
The third factor is equivalent to
Multiply these three factors:
The graph is negative since it goes down then up then down, so we have to switch all of the signs:
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Write the equation for the polynomial in this graph:
Write the equation for the polynomial in this graph:
The zeros for this polynomial are 
. That means that the factors are equal to zero when these values are plugged in.
or equivalently 
multiply both sides by 4
the first factor is 
multiply both sides by 3
the second factor is 
the third factor is 
Multiply the three factors:
The zeros for this polynomial are
Multiply the three factors:
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Write the equation for the polynomial in the graph:
Write the equation for the polynomial in the graph:
The zeros of the polynomial are 
. That means that the factors equal zero when these values are plugged in.
The first factor is 
or equivalently 
multiply both sides by 5:
The second and third factors are 
and 
Multiply:
Because the graph goes down-up-down instead of the standard up-down-up, the graph is negative, so change all of the signs:
The zeros of the polynomial are
The first factor is
The second and third factors are
Multiply:
Because the graph goes down-up-down instead of the standard up-down-up, the graph is negative, so change all of the signs:
Compare your answer with the correct one above