Investigation: Factored Form and x-Intercepts
Completion requirements
| Investigation |
Factored Form and x-Intercepts
Graph the quadratic function \(y =(x - 4)(x + 3)\), using your graphing calculator. What do you notice about the \(x\)-intercepts and the equation of the function?
Conclusion
In the Investigation, there seems to be a connection between the factored form of a quadratic function and the \(x\)-intercepts of the corresponding graph.
The \(x\)-intercepts are \(–3\) and \(4\). Looking at the equation of the function, the \(x\)-intercepts appear in the factored form of the quadratic, where
\(\begin{align}
y &= \left( {x - r} \right)\left( {x - s} \right) \\
r &= 4 \\
s &= -3 \\
y &= \left( {x - 4} \right)\left( {x + 3} \right) \end{align}\)
Using the function from the Investigation, what happens when \(x = 4 \)?
\(\begin{array}{l}
y = \left( {x - 4} \right)\left( {x + 3} \right) \\
y = \left( {4 - 4} \right)\left( {4 + 3} \right) \\
y = \left( 0 \right)\left( 7 \right) \end{array}\)
Notice that the first binomial becomes zero, and anything multiplied by zero is zero! The same thing happens if you were to solve for \(y\) when \(x = –3\). This time, the second binomial will be zero. This is why the \(x\)-intercepts of the graph of a function corresponds to the zeros of the function.
One method used to determine the zeros of a quadratic function is to factor the function. Once factored, calculate the values of \(x\) that will cause at least one factor to equal zero.