The graph of y = x^2 + 2x – 3 is a quadratic function that can be written in the form y = a(x – h)^2 + k, where a, h and k are constants. This form is called the vertex form of a quadratic function. The parent function of y = x^2 + 2x – 3 is y = x^2, which is also a quadratic function with a = 1, h = 0 and k = 0.
Comparing the graphs of y = x^2 + 2x – 3 and y = x^2
To compare the graphs of y = x^2 + 2x – 3 and y = x^2, we can use the vertex form to find the vertex, axis of symmetry, direction of opening and width of each parabola.
– The vertex of y = x^2 + 2x – 3 is (h, k) = (-1, -4), which means the parabola has been shifted left by 1 unit and down by 4 units from the origin. The vertex of y = x^2 is (0, 0), which is the origin.
– The axis of symmetry of y = x^2 + 2x – 3 is x = h = -1, which means the parabola is symmetric about the vertical line x = -1. The axis of symmetry of y = x^2 is x = 0, which is the y-axis.
– The direction of opening of y = x^2 + 2x – 3 is determined by the sign of a. Since a = 1, the parabola opens upward. The same is true for y = x^2, which also has a positive value of a.
– The width of y = x^2 + 2x – 3 is determined by the absolute value of a. Since |a| = 1, the parabola has the same width as y = x^2, which also has |a| = 1.
Here are the graphs of y = x^2 + 2x – 3 (in blue) and y = x^2 (in red) for comparison:
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graph {y=x^2+2x-3 [-10,10,-10,10]y=x^2 [-10,10,-10,10]}
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Finding the roots of y = x^2 + 2x – 3
Another way to compare the graphs of y = x^2 + 2x – 3 and y = x^2 is to find their roots or zeros, which are the values of x that make y equal to zero. To find the roots of y = x^2 + 2x – 3, we can use the quadratic formula:
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x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}
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where a = 1, b = 2 and c = -3. Plugging these values into the formula, we get:
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x=\frac{-2\pm\sqrt{4+12}}{2}
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Simplifying further, we get:
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x=\frac{-2\pm\sqrt{16}}{2}
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“`math
x=\frac{-2\pm4}{2}
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Therefore, the roots are:
“`math
x=\frac{-6}{2}=-3
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and
“`math
x=\frac{+6}{6}=1
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These are the points where the graph of y = x^2 + 2x – 3 crosses the x-axis. The graph of y = x^2 only has one root at (0,0), which is also its vertex.
Conclusion
The graph of y = x^2 + 2x – 3 is related to its parent function, y = x^2, by a horizontal and vertical translation and by having two distinct roots. Both graphs have the same shape, direction and width.
