The solutions of ax^2+bx+c=0 are the x-intercepts of the graph of y=ax^2+bx+c. Our equation is already written in standard form. Let's identify the function related to the equation.
Equation:&x^2-x-2=0
Related Function:&y=x^2-x-2
Graphing the Related Function
To draw the graph of the related function written in standard form, we must start by identifying the values of a, b, and c.
y=x^2-x-2 ⇔ y=1x^2+(- 1)x+(-2)
We can see that a=1, b=- 1, and c=-2. Now, we will follow four steps to graph the function.
The axis of symmetry is a vertical line with the equation x=- b2a. Since we already know the values of a and b, we can substitute them into the formula.
The axis of symmetry for the parabola is a vertical line with the equation x=0.5.
Calculating the Vertex
To calculate the vertex, we need to think of y as a function of x, y=f(x). We can write the expression for the vertex by stating the x- and y-coordinates in terms of a and b.
Vertex: ( - b/2a, f( - b/2a ) )
Note that the formula for the x-coordinate is the same as the formula for the axis of symmetry, which is x=0.5. Therefore, the x-coordinate of the vertex is also 0.5. To find the y-coordinate, we need to substitute 0.5 for x in the given equation.
We found the y-coordinate and now we know that the vertex is (0.5,- 2.25).
Identifying the y-intercept and its Reflection
The y-intercept of the graph for a quadratic function written in standard form is given by the value of c. The point where our graph intercepts the y-axis is (0,-2). Let's plot this point and its reflection across the axis of symmetry.
Connecting the Points
We can now draw the graph of the function. Since a=1, which is positive, the parabola will open upwards. Let's connect the three points with a smooth curve.
Finding the x-intercepts
Let's identify the x-intercepts of the graph of the related function.
We can see that the parabola intersects the x-axis twice. The points of intersection are ( -1,0) and ( 2,0). Therefore, the equation x^2-x-2=0 has two solutions, x= -1 and x= 2.
Checking Our Answer
Checking Our Solutions
We can check our solutions by substituting the values into the given equation. If our solutions are correct, the final result will be 0=0. Let's first check x=-1.
