Question

y = x ^ 2 + 2x - 24 i need to know the points at the x-intercept ( , ) and ( , ) points at the y intercept: ( , )and i’d it’s a min or max

Answer 1

Question : the function in the question is given below as

$y=x^2+2x-24$

Step 1: Calculate the x-intercept

The x-intercept for any curve is the value of the x coordinate of the point where the graph cuts the x-axis, or we can say that the x-intercept is the value of the x coordinate of a point where the value of the y coordinate is equal to zero.

Equating the equation above to zero (0)

$\begin{gathered} y=x^2+2x-24 \\ x^2+2x-24=0 \end{gathered}$

Solving using the quadratic formula below,

$x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}$

The general formula of a quadratic equation is given below

$ax^2+bx+c=0$

By comparing the coefficient, we will have the values to be

$\begin{gathered} a=1 \\ b=2 \\ c=-24 \end{gathered}$

Step 2: Substitute the values into the quadratic formula to get the values of x

$\begin{gathered} x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ x=\frac{-2\pm\sqrt[]{2^2-(4\times1\times-24)}}{2\times1} \\ x=\frac{-2\pm\sqrt[]{4^{}+96}}{2} \\ x=\frac{-2\pm\sqrt[]{100}}{2} \\ x=\frac{-2\pm10}{2} \\ x=\frac{-2+10}{2}\text{ or }x=\frac{-2-10}{2} \\ x=\frac{8}{2}\text{ or x=}\frac{\text{-12}}{2} \\ x=4\text{ or x=-6} \end{gathered}$

Hence,

The x-intercepts are

$(-6,0)\text{ and }(4,0)$

x-intercepts are (-6,0) and (4,0)

Step 3: Calculate the coordinate of the y-intercept

The point where a line or curve crosses the y-axis of a graph.

In other words: find the value when x equals 0

$\begin{gathered} y=x^2+2x-24 \\ y=(0)^2+2(0)-24 \\ y=0+0-24 \\ y=-24 \end{gathered}$

Hence,

The y-intercept is

$(0,-24)$

The y-intercept is (0,-24)

Below is the graph of the function on the question with its x-intercepts and y-intercepts

Step 4: Determine if the graph is minimum or maximum

The first step is to determine whether your equation gives a maximum or minimum. This can be done by looking at the x^2 term. If this term is positive, the vertex point will be a minimum; if it is negative, the vertex will be a maximum.

The coefficient of the x^2 term is a positive 1

Hence,

The equation

$y=x^2+2x-24\text{ }$

is a minimum quadratic graph