Practice Vertex and Axis of Symmetry in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The vertex of a parabola is the point where it reaches its maximum or minimum value. The axis of symmetry is the vertical line that passes through the vertex, dividing the parabola into two mirror-image halves.

Fold the parabola along the axis of symmetry and both halves match perfectly. The vertex is at the fold—the very bottom of a U-shaped parabola or the very top of an upside-down one. It is the point where the function changes direction.

Showing a random 20 of 50 problems.

Example 1

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The points (−1,3)(-1, 3) and (5,3)(5, 3) lie on a parabola. Find the vertex xx-coordinate.

Example 2

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Is the vertex of y=−2(x−3)2+10y = -2(x-3)^2 + 10 a maximum or minimum?

Example 3

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If the vertex of y=x2+bx+cy = x^2 + bx + c is (4,−3)(4, -3), find bb and cc.

Example 4

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Read the vertex of y=(x−6)2+2y = (x-6)^2 + 2.

Example 5

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Find the vertex of y=x2+6x+5y = x^2 + 6x + 5.

Example 6

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Find the axis of symmetry of y=x2−10x+7y = x^2 - 10x + 7.

Example 7

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A parabola passes through (1,0)(1, 0), (5,0)(5, 0), and (3,8)(3, 8). Find its equation in vertex form.

Example 8

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True or false: every parabola of the form y=ax2+bx+cy = ax^2 + bx + c has exactly one axis of symmetry.

Example 9

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What is the axis of symmetry of y=x2y = x^2?

Example 10

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Find the vertex xx-coordinate of y=2x2−12x+5y = 2x^2 - 12x + 5.

Example 11

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For what value(s) of kk does the parabola y=x2+kx+4y = x^2 + kx + 4 have its vertex on the line y=xy = x?

Example 12

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Two points (1,4)(1, 4) and (7,4)(7, 4) lie on a parabola. Find the axis of symmetry.

Example 13

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Does y=−x2+2x+5y = -x^2 + 2x + 5 have a maximum or minimum at its vertex?

Example 14

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A parabola has xx-intercepts −2-2 and 88. Find the axis of symmetry.

Example 15

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If f(2)=7f(2) = 7 and the axis of symmetry is x=5x = 5, find f(8)f(8).

Example 16

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A ball's height in feet is h(t)=−16t2+64t+5h(t) = -16t^2 + 64t + 5. When does it reach maximum height, and how high?

Example 17

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A farmer has 120120 m of fence to enclose a rectangular pen with one side along a river (no fence needed on that side). Find dimensions that maximize area.

Example 18

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Find the vertex of y=−(x−4)2+9y = -(x-4)^2 + 9.

Example 19

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Is the axis of symmetry a line or a point?

Example 20

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Find the axis of symmetry of y=x2+8x+1y = x^2 + 8x + 1.
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