Math Problem Statement

y=1/4x² +x-6 graph these equationns identify the axis of symmetry vertex and y intercept

Solution

To analyze and graph the quadratic equation y=14x2+x6y = \frac{1}{4}x^2 + x - 6, we need to determine the axis of symmetry, vertex, and y-intercept.


Step 1: Rewrite the equation in standard form

The given equation is already in the standard form: y=14x2+x6y = \frac{1}{4}x^2 + x - 6

Here:

  • a=14a = \frac{1}{4}
  • b=1b = 1
  • c=6c = -6

Step 2: Axis of Symmetry

The axis of symmetry for a parabola y=ax2+bx+cy = ax^2 + bx + c is given by: x=b2ax = -\frac{b}{2a}

Substitute a=14a = \frac{1}{4} and b=1b = 1: x=1214=112=2x = -\frac{1}{2 \cdot \frac{1}{4}} = -\frac{1}{\frac{1}{2}} = -2

Axis of Symmetry: x=2x = -2


Step 3: Vertex

To find the vertex, substitute x=2x = -2 into the equation to find yy: y=14(2)2+(2)6y = \frac{1}{4}(-2)^2 + (-2) - 6 y=14(4)26=126=7y = \frac{1}{4}(4) - 2 - 6 = 1 - 2 - 6 = -7

Vertex: (2,7)(-2, -7)


Step 4: Y-Intercept

The y-intercept occurs when x=0x = 0. Substitute x=0x = 0 into the equation: y=14(0)2+06=6y = \frac{1}{4}(0)^2 + 0 - 6 = -6

Y-Intercept: (0,6)(0, -6)


Step 5: Graphing

  1. Key Points:

    • Vertex: (2,7)(-2, -7)
    • Y-intercept: (0,6)(0, -6)
    • Symmetric Point: Reflect the y-intercept across the axis of symmetry at (4,6)(-4, -6).
  2. Plot the Points: Use x=3,1,1,4x = -3, -1, 1, -4 to calculate additional points if needed.

  3. Shape: The parabola opens upwards because a=14>0a = \frac{1}{4} > 0.


Summary of Properties:

  • Axis of Symmetry: x=2x = -2
  • Vertex: (2,7)(-2, -7)
  • Y-Intercept: (0,6)(0, -6)
  • Direction: Opens upwards

Would you like a plotted graph for better visualization?


Follow-Up Questions:

  1. What additional points would you like to calculate for the parabola?
  2. Do you need assistance with symmetry calculations for graphs?
  3. Should we explore how changing a,b,ca, b, c values affect the parabola's shape and position?
  4. Would you like a step-by-step guide on completing the square for vertex form?
  5. How does the vertex relate to the maximum or minimum value of the function?

Tip: Always check symmetry to simplify graphing quadratics!

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Graphing Parabolas

Formulas

Axis of symmetry formula: x = -b / 2a
Vertex form: y = ax^2 + bx + c
Y-intercept: c in y = ax^2 + bx + c

Theorems

Properties of Parabolas
Symmetry of Quadratic Functions

Suitable Grade Level

Grades 9-11