The Graph of y = x^2 + c
The graph of y = x^2 + c is a Grade 7 math skill from Yoshiwara Intermediate Algebra exploring how adding a constant c to the basic quadratic y = x^2 shifts the parabola vertically. Adding a positive c shifts up; adding a negative c shifts down; the vertex moves to (0, c).
Key Concepts
Property Compared to the graph of $y = x^2$, the graph of $y = x^2 + c$.
is shifted upward by $c$ units if $c 0$. is shifted downward by $|c|$ units if $c < 0$.
Examples The graph of $y = x^2 + 5$ is the basic parabola shifted 5 units up. Its vertex is located at $(0, 5)$. The graph of $y = x^2 3$ is the basic parabola shifted 3 units down. Its vertex is located at $(0, 3)$. The graph of $y = x^2 + 1$ is an upside down parabola that has been shifted 1 unit up. Its vertex is at $(0, 1).
Common Questions
How does adding c to y = x^2 change the graph?
Adding c shifts the entire parabola vertically. y = x^2 + 3 shifts the vertex up to (0, 3); y = x^2 - 2 shifts it down to (0, -2).
What is the vertex of y = x^2 + c?
The vertex is (0, c), directly on the y-axis at height c.
Does the shape of the parabola change when you add c?
No. The parabola has the same width and opens in the same direction. Only its vertical position changes.
How do you graph y = x^2 - 4?
Start from the basic y = x^2 parabola and slide it down 4 units so the vertex is at (0, -4).