Question

How do you find x-intercept, coordinates of vertex for parabola \[y={{x}^{2}}-4x-12\]?

Answer 1

Hint: To find the x-intercept of the given equation, you need to put the value of ‘y’ equal to zero and solve for the value of ‘x’ and you will get the x-intercept. To find the vertex of parabola, you need to find the derivative of the given quadratic equation and then equating it with equal to 0, then simplify further you will get the x-coordinate of the vertex and later putting it into the given quadratic equation you will get the y-coordinate of the vertex.

Complete step-by-step solution:
We have given that,
\[\Rightarrow y={{x}^{2}}-4x-12\]
Finding the x-intercept;
Putting y = 0,
We have,
\[\Rightarrow y={{x}^{2}}-4x-12\]
\[\Rightarrow 0={{x}^{2}}-4x-12\]
We get the standard quadratic equation.
Rewrite the equation as,
\[\Rightarrow {{x}^{2}}-4x-12=0\]
Splitting the middle term, we obtain
\[\Rightarrow {{x}^{2}}-6x+2x-12=0\]
Taking out common factors by making the pairs, we get
\[\Rightarrow x\left( x-6 \right)+2\left( x-6 \right)=0\]
Taking away common factors,
\[\Rightarrow \left( x+2 \right)\left( x-6 \right)=0\]
Equating each common factor equals to 0, we get
\[\Rightarrow x+2=0\ and\ x-6=0\]
Solving for the value of ‘x’, we get
\[\Rightarrow x=-2,6\]
So the x-intercept is at (0, 6) and (0, -2).
Now,
Finding the vertex;
We have,
\[\Rightarrow y={{x}^{2}}-4x-12\]
Differentiating it with respect to ‘x’, we get
\[\Rightarrow \dfrac{dy}{dx}=2x-4\]
Equating the result with 0, we obtain
\[\Rightarrow 2x-4=0\]
\[\Rightarrow 2x=4\]
\[\Rightarrow x=2\]
Putting the value of x = 2 in the given equation,
\[\Rightarrow y={{x}^{2}}-4x-12\]
\[\Rightarrow y={{2}^{2}}-4\left( 2 \right)-12\]
\[\Rightarrow y=4-8-12=-16\]
\[\Rightarrow y=-16\]
Therefore the vertex of the parabola is at (2, -16).

Note: We have an alternative method of find the vertex of the given parabola by using the vertex form of the equation; i.e. \[y=a{{\left( x-h \right)}^{2}}+k\] where, \[a\] equals to the coefficient of \[{{x}^{2}}\], ‘h’ is the x-coordinate of the vertex, ‘k’ is the y-coordinate of the vertex. Students can use any of the methods to find the vertex of the parabola as answers from both the methods will get the same.