Plot the following and connect the dots:
y-intercept: (0, -3)
x-intercepts: (1, 0) and (-3, 0)
vertex: (-1, -4)
Step-by-step explanation:
To graph an equation, you need some points that are on the graph.
y = x² + 2x - 3
This equation is given in standard form, y = ax² + bx + c.
"c" is the y-intercept, where the graph hits the y-axis.
Since c = -3, one point is:
(0, -3)
a = 1; b = 2; c = -3
y = x² + 2x - 3
Since a = 1, we can factor by breaking down "b".
Find two numbers that add to get "b", and also multiply to get "c".
? + ? = 2
? × ? = -3
Try 3 and -1:
3 + -1 = 2 Correct
3 × -1 = -3 Correct
Split the "b" value into p and q. Use group factoring.
y = x² - x + 3x - 3
Group terms that looks alike.
y = (x² - x) + (3x - 3)
Collect another pair of like terms.
y = x(x - 1) + 3(x - 1)
y = (x - 1)(x + 3) This is fully factored. Use this to find the roots.
The roots are the x-intercepts, when y = 0.
0 = (x - 1)(x + 3)
Split the factors and equate them to 0.
0 = x - 1 0 = x + 3
x = 1 x = -3
(1, 0) (-3, 0)
We also need the vertex. Use complete the square.
y = x² + 2x - 3
y = (x² + 2x) - 3 Divide the "b" term by 2, and square. (2/2)² = 1
y = (x² + 2x + 1 - 1) - 3 Add and subtract 1.
y = (x² + 2x + 1) - 1 - 3 Take out the negative.
y = (x² + 2x + 1) - 4 Simplify
Now factor the bracketed expression, just like before.
x² + 2x + 1
? + ? = 2
? × ? = 1
Try 1 and 1:
1 + 1 = 2
1 × 1 = 1
The factored expression is: (x + 1)(x + 1) = (x + 1)²
Put it back into the equation for the graph:
y = (x + 1)² - 4
This is vertex form: y = a(x - h)² + k, where the vertex is (h, k).
(-1, -4)
See the graph below, the blue is the plotted points.
