Graphing Quadratic Equations
Graphing Quadratic Equations
Quadratic equations are second-degree polynomials; the highest power on the dependent variable is 2. The graph of a quadratic function is a parabola, which is U-shaped and has three important components. The vertex is where the graph changes direction. The axis of symmetry is the vertical line that cuts the graph into two equal halves. The axis of symmetry always passes through the vertex. The zeros or roots of the quadratic are the x-intercepts of the graph.
Quadratic equations can be expressed in standard form or vertex form.
Standard form: y = ax2 + bx + c | Vertex form: y = a(x – h)2 + k |
|---|---|
Axis of symmetry: \(x = −\frac{b}{2a}\) Vertex: \((−\frac{b}{2a}, f (−\frac{b}{2a}))\) | Vertex: (h, k) Axis of symmetry: x = h |
In both equations, the sign of a determines which direction the parabola opens: if a is positive, it opens upward; if a is negative, it opens downward.
Equations in vertex form can be converted to standard form by squaring (x – h) using FOIL, distributing the a, adding k, and simplifying the result.
Write y = 2(x – 6)2 – 14 in standard form.
Square the (x – h) component.
y = 2(x – 6)2 – 14
y = 2(x – 6)(x – 6) – 14
y = 2(x2 – 12x + 36) – 14
Distribute the a and simplify.
y = 2x2 – 24x + 72 – 14
y = 2x2 – 24x + 58
- Move c to the left side of the equation by subtracting it from both sides.
- Divide the entire equation by a (so the coefficient of x2 is 1).
- Take half of the coefficient of x, square it, and add the quantity to both sides of the equation.
- Convert the right side of the equation to a perfect binomial squared, (x + m)2.
- Isolate y to put the equation in proper vertex form.
Write y = –3x 2 + 24x – 27 in vertex form.
Move c to the other side of the equation.
\(y = −3x^{2} + 24x\ –\ 27\)
\(y\ +\ 27 = −3x^{2} + 24x\)
\(Divide\ by\ a = −3\)
\(−\frac{y}{3}\ −\ 9 = x^{2}\ − 8x\)
Take half of the new b, square it, and add that quantity to both sides:
\(\frac{1}{2}(−8) = −4\ and\ (−4)^{2} = 16\)
\(−\frac{y}{3}\ −\ 9\ +\ 16 = x^{2}\ −\ 8x\ +\ 16\)
Write the right side as a squared binomial and simplify.
\(−\frac{y}{3}\ +\ 7\ = (x\ −\ 4)^{2}\)
Rewrite the equation in vertex form.
\(y = −3(x\ −\ 4)^{2}\ 21\)