Q. Plot the given parabola on the axes. Plot the roots, the vertex, and two other points. Equation: \(y=x^2+2x-3\)

Problem

Plot the parabola given by the equation \(y = x^{2} + 2x – 3\). On the axes, mark the roots, the vertex, and two other points.

Step-by-step solution and instructions for plotting

1. Write the function clearly.The parabola is given by

   \(y = x^{2} + 2x – 3\).

2. Find the roots (x-intercepts) by factoring.Attempt to factor the quadratic \(x^{2} + 2x – 3\). Look for two numbers that multiply to \(-3\) and add to \(2\). Those numbers are \(3\) and \(-1\). Thus

   \(x^{2} + 2x – 3 = (x + 3)(x – 1)\).

   Set each factor equal to zero to get the roots:

   \(x + 3 = 0 \Rightarrow x = -3\)

   \(x – 1 = 0 \Rightarrow x = 1\)

   So the roots (points where the parabola crosses the x-axis) are

   \((-3, 0)\) and \((1, 0)\).

3. Find the vertex.For a parabola \(y = ax^{2} + bx + c\) the x-coordinate of the vertex is \(x_{v} = -\dfrac{b}{2a}\). Here \(a = 1\) and \(b = 2\), so

   \(x_{v} = -\dfrac{2}{2 \cdot 1} = -1\).

   Now compute the y-coordinate by evaluating the function at \(x = -1\):

   \(y_{v} = (-1)^{2} + 2(-1) – 3 = 1 – 2 – 3 = -4\).

   Thus the vertex is at

   \((-1, -4)\).

   Because \(a = 1 > 0\), the parabola opens upward.

4. Choose two other points (nice symmetric points) to help shape the graph.Choose x-values symmetric about the axis of symmetry \(x = -1\). Two convenient choices are \(x = 0\) and \(x = -2\), which are each one unit from \(x = -1\).

   Evaluate the function at those x-values:

   For \(x = 0\): \(y = 0^{2} + 2(0) – 3 = -3\). So the point is \((0, -3)\).

   For \(x = -2\): \(y = (-2)^{2} + 2(-2) – 3 = 4 – 4 – 3 = -3\). So the point is \((-2, -3)\).

   These two points are symmetric about the axis \(x = -1\) and lie above the vertex.

5. Summary of key points to plot.
   • Roots: \((-3, 0)\) and \((1, 0)\).
   • Vertex: \((-1, -4)\).
   • Two other points: \((-2, -3)\) and \((0, -3)\).

   A small table of values (useful for plotting):

   x	y = x^{2} + 2x – 3	Point
   -3	0	(-3, 0)
   -2	-3	(-2, -3)
   -1	-4	(-1, -4) vertex
   0	-3	(0, -3)
   1	0	(1, 0)

6. How to draw the graph on axes (step-by-step plotting instructions).
   1. Draw a pair of perpendicular axes. Mark integer ticks on both axes, at least from x = -4 to x = 2 and y = -5 to y = 2.
   2. Plot the roots at (-3, 0) and (1, 0). These are points on the x-axis.
   3. Plot the vertex at (-1, -4). Mark it clearly (this is the lowest point because the parabola opens upward).
   4. Plot the two symmetric points (-2, -3) and (0, -3). They show the parabola’s shape near the vertex.
   5. Draw a smooth U-shaped curve through these points, making sure it passes through the roots, goes down to the vertex, and is symmetric about the vertical line x = -1.

   That completes the plotting instructions. The resulting graph is the parabola with vertex at (-1, -4), crossing the x-axis at -3 and 1, and passing through (-2, -3) and (0, -3).