10.5 Graphing Quadratic Equations in Two Variables

Parabolas are U-shaped curves that show up whenever you graph a quadratic equation. Understanding their key features (vertex, axis of symmetry, intercepts) lets you quickly sketch a graph and interpret what the equation is telling you.

This section covers how to identify those features from an equation in standard form and how to use them to produce an accurate graph.

Understanding Parabolas and Their Key Features

Features of parabolas

A parabola is a U-shaped curve that's symmetrical about a vertical line called the axis of symmetry. Every parabola either opens upward or downward, and the leading coefficient $a$ in the standard form $y = ax^2 + bx + c$ controls which way:

• If $a > 0$, the parabola opens upward (like a cup). As $x$ gets large in either direction, $y$ increases.
• If $a < 0$, the parabola opens downward (like an upside-down cup). As $x$ gets large in either direction, $y$ decreases.

The size of $a$ also affects how wide or narrow the parabola is. A larger absolute value of $a$ (like $a = 3$) makes the parabola narrower, while a smaller absolute value (like $a = 0.5$) makes it wider.

Vertex and axis of symmetry

The vertex is the turning point of the parabola. If the parabola opens upward, the vertex is the lowest point (minimum). If it opens downward, the vertex is the highest point (maximum).

To find the vertex from $y = ax^2 + bx + c$:

1. Calculate the x-coordinate: $x = -\frac{b}{2a}$
2. Plug that x-value back into the original equation to get the y-coordinate.
3. The vertex is the point $\left(-\frac{b}{2a},\; y\right)$.

For example, with $y = 2x^2 - 8x + 3$: the x-coordinate is $x = -\frac{-8}{2(2)} = 2$. Substituting back: $y = 2(2)^2 - 8(2) + 3 = 8 - 16 + 3 = -5$. So the vertex is $(2, -5)$.

The axis of symmetry is the vertical line $x = -\frac{b}{2a}$, which passes straight through the vertex. It divides the parabola into two mirror-image halves. In the example above, the axis of symmetry is $x = 2$.

Intercepts of parabolas

x-intercepts are where the parabola crosses the x-axis (where $y = 0$). To find them, set $y = 0$ in the equation and solve the resulting quadratic. You can use factoring, the quadratic formula, or completing the square.

A parabola can have:

• Two x-intercepts (the curve crosses the x-axis twice)
• One x-intercept (the vertex sits right on the x-axis)
• Zero x-intercepts (the curve never reaches the x-axis)

These x-intercepts are also called the roots or zeros of the quadratic equation.

y-intercept is where the parabola crosses the y-axis (where $x = 0$). To find it, substitute $x = 0$ into the equation. In standard form $y = ax^2 + bx + c$, this always gives $y = c$. So the y-intercept is the point $(0, c)$. Every parabola has exactly one y-intercept.

Graphing quadratic equations

Here's the process for graphing $y = ax^2 + bx + c$:

1. Find the vertex. Use $x = -\frac{b}{2a}$, then plug that x-value back in to get y.
2. Draw the axis of symmetry. Sketch a dashed vertical line through the vertex.
3. Find the y-intercept. Set $x = 0$; the y-intercept is $(0, c)$.
4. Find x-intercepts (if they exist). Set $y = 0$ and solve the quadratic equation.
5. Plot a couple of extra points. Choose an x-value on one side of the vertex, calculate y, then use symmetry to plot the mirror point on the other side.
6. Connect the points with a smooth curve. Make sure the curve reflects the correct direction (up or down) based on the sign of $a$.

The symmetry property is your best friend here. If you know a point on one side of the axis of symmetry, you automatically know the corresponding point on the other side.

Applications of quadratic graphs

Quadratic equations model many real-world situations:

• Projectile motion: The height of a ball thrown upward over time follows a downward-opening parabola. The vertex gives the maximum height.
• Area optimization: If you have a fixed amount of fencing and want to enclose the largest rectangular area, the problem reduces to a quadratic. The vertex tells you the dimensions that maximize area.
• Profit maximization: Revenue or profit as a function of production level often forms a parabola. The vertex identifies the production level that yields the highest profit.

To solve these kinds of problems:

1. Identify or set up the quadratic equation that models the situation.
2. Find the vertex, since it represents the maximum or minimum value.
3. Interpret the vertex coordinates in context. The x-coordinate tells you where the max/min occurs (time, quantity, etc.), and the y-coordinate tells you what that max/min value is.

Quadratic Functions and Polynomials

Quadratic equations are polynomial functions of degree 2. A polynomial is an expression built from variables and coefficients using only addition, subtraction, and multiplication (no division by variables, no square roots of variables).

As a function, every x-value you plug in produces exactly one y-value. That's why the graph passes the vertical line test: any vertical line crosses the parabola at most once. Graphing quadratic functions on a coordinate plane helps you visualize how the output changes across different input values and across different quadrants.