The Graph of y = ax^2
The graph of y = ax^2 is a Grade 7 math skill from Yoshiwara Intermediate Algebra exploring how the coefficient a affects the shape of a basic parabola. When a > 1, the parabola is narrower than y = x^2; when 0 < a < 1, it is wider; and when a < 0, it opens downward.
Key Concepts
Property The parabola opens upward if $a 0$. The parabola opens downward if $a < 0$. The magnitude of $a$ determines how wide or narrow the parabola is. The vertex, the $x$ intercepts, and the $y$ intercept all coincide at the origin.
Examples The graph of $y = 4x^2$ opens upward and is narrower than the basic parabola $y=x^2$. It passes through the points $( 1, 4)$ and $(1, 4)$. The graph of $y = \frac{1}{3}x^2$ opens downward and is wider than the basic parabola. It passes through the points $( 3, 3)$ and $(3, 3)$. The graph of $y = 0.25x^2$ opens upward and is wider than the basic parabola. It passes through the points $( 2, 1)$ and $(2, 1).
Explanation The coefficient 'a' acts like a stretch factor that controls the parabola's direction and width. A positive 'a' makes it open up, while a negative 'a' flips it upside down. A larger absolute value of 'a' creates a narrower parabola.
Common Questions
What does the graph of y = ax^2 look like?
The graph is a parabola with vertex at the origin (0,0). If a > 0, it opens upward; if a < 0, it opens downward.
How does the value of a affect the width of the parabola?
Larger |a| makes the parabola narrower; smaller |a| makes it wider. y = 3x^2 is narrower than y = x^2, while y = (1/3)x^2 is wider.
What is the vertex and axis of symmetry of y = ax^2?
The vertex is (0, 0) and the axis of symmetry is the y-axis (x = 0).
How is y = ax^2 different from y = x^2 + c?
y = ax^2 scales the parabola vertically (stretches/compresses), while y = x^2 + c shifts it up or down.