Graphing quadratic equations
Graphing quadratic equations is a Grade 7 math skill from Yoshiwara Intermediate Algebra where students plot parabolas by identifying the vertex, axis of symmetry, y-intercept, and x-intercepts. Understanding the shape and key features of the parabola enables accurate sketching.
Key Concepts
Property The graph of a quadratic equation $y = ax^2 + bx + c$ is called a parabola . For the graph of $y = ax^2$: The parabola opens upward if $a 0$ and downward if $a < 0$. The magnitude of $a$ determines how wide or narrow the parabola is. For the graph of $y = x^2 + c$: The graph is shifted upward by $c$ units if $c 0$ and downward if $c < 0$. For the graph of $y = ax^2 + bx + c$, the $x$ coordinate of the vertex is $x v = \frac{ b}{2a}$.
Examples For the parabola $y = x^2 6x + 5$, the vertex's x coordinate is $x v = \frac{ ( 6)}{2(1)} = 3$. To find the y coordinate, plug $x=3$ back in: $y = (3)^2 6(3) + 5 = 9 18 + 5 = 4$. The vertex is at $(3, 4)$.
The graph of $y = 3x^2$ opens downward because $a= 3$ is less than 0. It is also narrower than the basic parabola $y=x^2$ because the magnitude of $a$ is greater than 1.
Common Questions
How do you graph a quadratic equation?
Find the vertex using x = -b/(2a), calculate the y-coordinate, identify x-intercepts by solving y = 0, find the y-intercept by setting x = 0, then plot and connect the points.
What is the axis of symmetry of a parabola?
The axis of symmetry is the vertical line x = -b/(2a), passing through the vertex. The parabola is a mirror image on both sides.
How many x-intercepts can a parabola have?
A parabola can have 0, 1, or 2 x-intercepts depending on whether the discriminant is negative, zero, or positive.
What does a wider or narrower parabola indicate?
A smaller |a| value makes the parabola wider (flatter), while a larger |a| makes it narrower (steeper).