How do you determine the characteristics of a quadratic function?

Concavity: Is it a Smile or a Frown? This is all about whether the parabola opens upwards or downwards. Remember that ‘a’ from our equation? That’s the key. If ‘a’ is positive (a > 0), the parabola opens upwards, like a happy smile. This means the vertex (we’ll get to that in a sec) is the lowest point on the graph, the minimum value. On the flip side, if ‘a’ is negative (a < 0), the parabola opens downwards, like a sad frown. Now the vertex is the highest point, the maximum value. I always remember it this way: positive vibes, upward smile!

The Vertex: The Turning Point. The vertex is the most important point on the parabola. It’s where the curve changes direction, kind of like that moment when you realize you’re going the wrong way on a road trip. It’s either the lowest point (if it’s a smile) or the highest point (if it’s a frown). And there’s a special way to write the equation that makes the vertex super obvious:

f(x) = a(x – h)² + k

This is called vertex form, and the vertex is simply the point (h, k). Easy peasy! If you’ve only got the standard form, don’t worry. You can find the x-coordinate of the vertex using the formula h = -b/(2a). Then, just plug that value back into the original equation to find the y-coordinate, k = f(h).

Axis of Symmetry: The Mirror Image. Imagine drawing a line straight down the middle of the parabola, through the vertex. That’s the axis of symmetry. It’s like a mirror; whatever’s on one side of the parabola is exactly the same on the other. The equation for this line is simply x = h, where ‘h’ is the x-coordinate of the vertex.

X-Intercepts: Where it Crosses the Line. The x-intercepts are the points where the parabola crosses the x-axis. These are also called roots or zeros, because they’re the values of ‘x’ that make the function equal to zero (f(x) = 0). To find them, you’ve got a few options: factoring, using the quadratic formula, or even completing the square.

• Factored Form: Sometimes, you’ll see the quadratic written like this: f(x) = a(x – r)(x – s). In this case, ‘r’ and ‘s’ are the x-intercepts. Boom!

Y-Intercept: The Easy One. This is where the parabola crosses the y-axis. To find it, just plug in x = 0 into the standard form equation. You’ll get f(0) = c. So, the y-intercept is simply the point (0, c). That’s why standard form is sometimes useful.

Domain and Range: What’s Allowed, What’s Possible. The domain is all the possible ‘x’ values you can plug into the function. For quadratics, that’s always all real numbers. You can plug in anything you want! The range, however, is a bit trickier. It’s all the possible ‘y’ values that the function can output. This depends on whether the parabola opens up or down, and where the vertex is.

• If a > 0 (opens up), the range is all ‘y’ values greater than or equal to the y-coordinate of the vertex: k, ∞).
• If a < 0 (opens down), the range is all 'y' values less than or equal to the y-coordinate of the vertex: (-∞, k.

The Discriminant: Predicting the Roots. This is a sneaky little calculation that tells you how many x-intercepts the parabola has before you even try to find them. It’s part of the quadratic formula:

Δ = b² – 4ac

• If Δ > 0, you’ve got two distinct real roots (two x-intercepts).
• If Δ = 0, you’ve got one real root (the vertex touches the x-axis).
• If Δ < 0, you've got no real roots (the parabola doesn't cross the x-axis). These roots are complex numbers.

Algebraic Kung Fu:

• Factoring: Break down the equation to find those x-intercepts.
• Quadratic Formula: Your trusty fallback for finding x-intercepts.
• Completing the Square: Transform the equation into vertex form.
• The -b/(2a) Trick: Quickly find the x-coordinate of the vertex.
• Discriminant Detective Work: Determine the number of roots.

Graphical Sleuthing:

• Graphing Calculators & Software: Tools like Desmos and GeoGebra make visualizing the parabola a breeze.
• Eyeballing the Graph: Just look at the graph! The vertex, intercepts, and concavity are all right there.

• h = -b/(2a) = -(-8)/(2*2) = 2.
• k = f(2) = 2(2)² – 8(2) + 6 = -2.
• So, the vertex is (2, -2).

• Factor it: 2x² – 8x + 6 = 2(x² – 4x + 3) = 2(x – 1)(x – 3).
• Set f(x) = 0: 2(x – 1)(x – 3) = 0, so x = 1 or x = 3.
• X-intercepts: (1, 0) and (3, 0).