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.

Sketch the graph of $y = x^2 - 4$ and find its roots.

Answer: The roots of the equation $y = x^2 - 4$ are $x = 2$ and $x = -2$ . The graph is a parabola opening upwards with its vertex at $(0, -4)$ , intersecting the $x$ -axis at $(2, 0)$ and $(-2, 0)$ , and the $y$ -axis at $(0, -4)$ .

Step-by-step solution

Step 1: Identify the type of equation

The given equation is a quadratic equation of the form $y = ax^2 + bx + c$ . 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 $x^2 - 4$ 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 $y = ax^2 + bx + c$ , 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.

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Sketch the graph of y=x2−4y = x^2 - 4y=x2−4 and find its roots.