Video Transcript
Write the quadratic equation
represented by the graph shown. Give your answer in standard
form.
In this question, we are given the
graph of a quadratic curve, and we are asked to use this given graph to determine
the equation of the curve in standard form. We can start by recalling that the
standard form of a quadratic equation is the form 𝑦 equals 𝑎𝑥 squared plus 𝑏𝑥
plus 𝑐, where 𝑎, 𝑏, and 𝑐 are constants. This means that we need to find the
values of these constants using the graph. We can find the value of 𝑐 by
substituting 𝑥 equals zero into the quadratic equation. All of the terms on the right-hand
side of the equation, except the constant term, have a factor of zero. So, this simplifies to give 𝑦
equals 𝑐.
Therefore, when 𝑥 equals zero, we
have that 𝑦 is equal to 𝑐. In the graph, this will be the
𝑦-intercept. We see that this is at negative 16,
so 𝑐 equals negative 16. It’s worth noting that in general,
for a quadratic equation in standard form, the 𝑦-intercept is always the constant
term 𝑐. This means that we have shown that
the quadratic has an equation of the form 𝑦 equals 𝑎𝑥 squared plus 𝑏𝑥 minus
16. We can determine the values of 𝑎
and 𝑏 by substituting the coordinates of known points on the curve into this
equation.
From the given graph, we can see
that the 𝑥-intercepts are at negative eight and two. This means that the points with
coordinates negative eight, zero and two, zero lie on the curve. This means that the coordinates of
these points satisfy the equation of the quadratic. If we substitute 𝑥 equals two and
𝑦 equals zero into our equation of the curve, we will find an equation involving 𝑎
and 𝑏. We have zero equals 𝑎 multiplied
by two squared plus 𝑏 times two minus 16. We can now simplify the equation to
get zero equals four 𝑎 plus two 𝑏 minus 16. Then we add 16 to both sides of the
equation to obtain 16 equals four 𝑎 plus two 𝑏.
This is a linear equation in two
unknowns, so we cannot solve this equation for 𝑎 and 𝑏. Instead, let’s follow the same
process for our other 𝑥-intercept. This will give us two equations in
two unknowns, which we can solve simultaneously. Substituting 𝑦 equals zero and 𝑥
equals negative eight into the equation yields zero equals 𝑎 times negative eight
squared plus 𝑏 multiplied by negative eight minus 16. We can evaluate the coefficients to
get zero equals 64𝑎 minus eight 𝑏 minus 16. We can also note that all three
terms share a factor of eight. So, we can simplify the equation by
dividing both sides of the equation by eight. This gives us zero equals eight 𝑎
minus 𝑏 minus two. We can then add two to both sides
of the equation to obtain two equals eight 𝑎 minus 𝑏.
We now have two linear equations in
two unknowns. We can solve these as simultaneous
equations. Let’s clear some space and then
solve the equations to find the values of 𝑎 and 𝑏. There are many ways of solving
simultaneous equations, and we will only go through one of these. We note that the coefficients of 𝑏
in the two equations have opposite signs and that we can double the second equation
to obtain equal and opposite terms of positive and negative two 𝑏. Doubling the second equation gives
us four equals 16𝑎 minus two 𝑏. We can now add these two equations
together to eliminate the unknown 𝑏. Adding the two equations together
gives us 16 plus four equals four 𝑎 plus 16𝑎, and the 𝑏-terms cancel.
We can now simplify and solve the
equation. Simplifying gives us that 20 equals
20𝑎. And then dividing both sides of the
equation by 20 gives us 𝑎 equals one. We can now find the value of 𝑏 by
substituting 𝑎 equals one into either of the simultaneous equations. Substituting 𝑎 equals one into the
first simultaneous equation gives us 16 equals four times one plus two 𝑏. We can then simplify and solve for
𝑏. We have 16 equals four plus two
𝑏. We can subtract four from both
sides of the equation to obtain 12 equals two 𝑏. Finally, we can divide the equation
through by two to find 𝑏 equals six.
There are a few things we can do to
check our answer. First, we can see that the parabola
opens upwards. This means that the value of 𝑎
must be positive. Second, we can also substitute the
values of 𝑎 and 𝑏 into the second equation to verify that it is satisfied. We can now substitute 𝑎 equals
one, 𝑏 equals six, and 𝑐 equals negative 16 into the standard form of the equation
of a quadratic to get that the quadratic equation shown has equation 𝑦 equals 𝑥
squared plus six 𝑥 minus 16 in standard form.
