Sketch the graph of y=x2 - 4 and find its roots.
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Step-by-Step Solution
Step 1: Identify the type of equation
The given equation is a quadratic equation of the form y=ax2+bx+c. In this case, a=1, b=0, and c=−4. Since a>0, the parabola opens upwards.
Step 2: Find the roots (x-intercepts)
To find the roots, we set y=0 and solve for x. This means finding the points where the graph intersects the x -axis. We can factor the expression x2−4 as a difference of squares.
Step 3: Find the y-intercept
To find the y -intercept, we set x=0 and solve for y. This is the point where the graph intersects the y -axis.
Step 4: Identify the vertex
For a parabola y=ax2+bx+c, the x -coordinate of the vertex is given by the formula x=−b/(2a). Since b=0, the vertex is at x=0. Substituting x=0 into the equation gives the y -coordinate of the vertex.
Step 5: Sketch the graph
With the roots at (2,0) and (−2,0), the y -intercept at (0,−4), and the vertex at (0,−4), we can sketch the parabola. Since the parabola opens upwards, it will pass through these points.