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Recognize quadratic relations from graphs

From a coordinate graph, a quadratic relation is identified by its characteristic parabola: a single axis of symmetry, a vertex, and consistent curvature, opening upward when the leading coefficient is positive and downward when it is negative. The graph is interpreted through its vertex, axis of symmetry, y-intercept, and x-intercepts (zeros), while distinguishing it from linear, absolute-value, and other nonlinear graphs; this connects graphical recognition with solving and interpreting quadratic equations. The focus is on vertical parabolas of the form y=ax2+bx+cy=ax^2+bx+c, not rotated conics or more general quadratic curves.

Detailed Explanation: Recognize quadratic relations from graphs

A quadratic relation has a graph shaped like a parabola. To recognize one, check for:

  • One line of symmetry, called the axis of symmetry
  • One turning point, called the vertex
  • A smooth, consistent curve
  • An opening upward or downward

A vertical parabola can be written as

y=ax2+bx+cy=ax^2+bx+c

If a>0a>0, it opens upward. If a<0a<0, it opens downward.

Example: Suppose a graph contains the points

(−2,3),(−1,0),(0,−1),(1,0),(2,3).(-2,3),\quad (-1,0),\quad (0,-1),\quad (1,0),\quad (2,3).

Step 1: Look at the overall shape.
The graph decreases until it reaches (0,−1)(0,-1) and then increases. It has a smooth, U-shaped curve, so it could be a parabola.

Step 2: Find the vertex.
The lowest point is

(0,−1).(0,-1).

This is the vertex.

Step 3: Check for symmetry.
The points on either side of x=0x=0 match:

  • At x=−1x=-1 and x=1x=1, y=0y=0
  • At x=−2x=-2 and x=2x=2, y=3y=3

Therefore, the axis of symmetry is

x=0.x=0.

Step 4: Identify the intercepts.

  • The graph crosses the yy-axis at (0,−1)(0,-1), so the yy-intercept is −1-1.
  • The graph crosses the xx-axis at (−1,0)(-1,0) and (1,0)(1,0), so the zeros are x=−1x=-1 and x=1x=1.

Step 5: State the conclusion.
The graph is a quadratic relation because it is a smooth, U-shaped curve with one vertex and one axis of symmetry. It opens upward, so its leading coefficient is positive.

It matches the equation

y=x2−1.y=x^2-1.

A straight line is linear, and a V-shaped graph is an absolute-value relation. Neither has the smooth, curved shape and single vertex of this parabola.

Learn by doing: Recognize quadratic relations from graphs

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    Quadratic Discriminants - Equation to Graph

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