To solve a system of two functions of a single variable we look for values of the domain variable that give the same function value for each function. We see how this is done for a pair of quadratic functions.
Example 1
If $f(x)=x^2+1$f(x)=x2+1 and $g(x)=-x^2+x+4$g(x)=−x2+x+4, we want to find the value or values of $x$x such that $f(x)=g(x)$f(x)=g(x). We can begin by assuming the two functions have the same value, which means we can write
$x^2+1=-x^2+x+4$x2+1=−x2+x+4.
After collecting like terms, we have the quadratic equation $2x^2-x-3=0$2x2−x−3=0 and with the help of the quadratic formula or by writing the left-hand side in factored form $(2x-3)(x+1)=0$(2x−3)(x+1)=0, we see that the solutions are
$x=\frac{3}{2}$x=32 and $x=-1$x=−1
Then, we substitute these $x$x-values into either one of the original functions to find the corresponding function values. Thus,
$f\left(\frac{3}{2}\right)=\left(\frac{3}{2}\right)^2+1=\frac{13}{4}$f(32)=(32)2+1=134 and
$f\left(-1\right)=\left(-1\right)^2+1=2$f(−1)=(−1)2+1=2
We conclude that the points of intersection of the graphs of the two functions are
$(-1,2)$(−1,2) and
$\left(\frac{3}{2},\frac{13}{4}\right)$(32,134)
The graphs are shown below.
A system of quadratic functions may be given in the form of a pair of equations of the form $y=ax^2+bx+c$y=ax2+bx+c. It is possible for a system of two quadratic functions to have no points in common, one common point or two.
Example 2
Two quadratic functions are given by
| $y$y | $=$= | $x^2+x-1$x2+x−1 |
| $y$y | $=$= | $4x^2+3x-\frac{2}{3}$4x2+3x−23 |
Their graphs are shown below.
The graphs appear to intersect in one point. We check this algebraically.
We can eliminate $y$y from the two equations by subtracting the first equation from the second. This gives the quadratic equation $3x^2+2x+\frac{1}{3}=0$3x2+2x+13=0. According to the quadratic formula, this has the single solution $x=-\frac{1}{3}$x=−13. At this value of $x$x, the corresponding $y$y-value in the original equations is $-\frac{11}{9}$−119.
Example 3
In business, supply and demand functions are ways of expressing the supply or demand of a commodity as a function of its unit price. Market equilibrium is the term that describes when the supply is equal to the demand. Consider the supply and demand functions for a new power drill below, where $y$y is in dollars and $x$x is in thousands of units supplied.
| Supply function | $y=4x^2$y=4x2 |
| Demand function | $y=-6x+5x^2$y=−6x+5x2 |
We want to find the equilibrium quantity and the corresponding price by solving the system of equations. Start by solving for $x$x.
Find the value of $y$y when $x=0$x=0.
Find the value of $y$y when $x=6$x=6.
Hence, the equilibrium quantity occurs when $\editable{}$ thousand units are being produced, which has a corresponding price of $\editable{}$ dollars per unit.
Example 4
Consider the system of equations
$y=x^2+4x+1$y=x2+4x+1
$y=-x^2+7$y=−x2+7
Shade the region contained inside the graphs (including the boundaries).
Loading Graph...What are the coordinates of the points of intersection?
Write the coordinates of both points on the same line separated by a comma.
Example 5
Consider the system of equations
$y=2x^2-3x-5$y=2x2−3x−5
$y=x^2-2x+1$y=x2−2x+1
What are the coordinates of the points of intersection?
Write the coordinates of both points on the same line separated by a comma.