Graph quadratic functions, find the vertex and axis of symmetry, identify zeros and the y-intercept, and connect standard, factored, and vertex forms to the parabola's features.

What this topic is asking

The Functions category requires you to graph and analyze quadratic functions (the F-IF and A-SSE standards). On the Grade 10 MCAS you find the vertex and axis of symmetry, identify the zeros and y-intercept, state which way the parabola opens, and connect the three algebraic forms to these features. This builds directly on solving quadratics, and it is a reliable source of constructed-response questions.

The shape and direction of a parabola

Every quadratic graphs as a symmetric U-shaped parabola. The leading coefficient $a$ controls two things:

• Direction. $a > 0$ opens upward (a valley with a minimum); $a < 0$ opens downward (a hill with a maximum).
• Width. A larger $|a|$ makes a narrower parabola; a smaller $|a|$ makes a wider one.

The MCAS often asks whether a function has a maximum or a minimum; the sign of $a$ answers it without any computation.

Axis of symmetry and vertex

The axis of symmetry is the vertical line through the vertex that mirrors the parabola. From standard form it is

This formula is not on the reference sheet, so memorize it, and note the leading minus sign. To find the vertex, compute the axis $x$-value, then substitute it back into the function to get the $y$-value. The vertex is the maximum or minimum point.

Zeros and the y-intercept

The zeros (x-intercepts, roots) are where $y = 0$, found by factoring or the quadratic formula. A parabola can have two, one (a double root, where the vertex touches the x-axis), or zero real x-intercepts (when the discriminant is negative).

The y-intercept is where $x = 0$, which in standard form is simply the constant $c$. So for $y = x^2 - 6x + 5$, the y-intercept is $5$ at the point $(0, 5)$.

A useful shortcut: the vertex's $x$-coordinate is the average of the two zeros, because the parabola is symmetric. If the zeros are $1$ and $5$, the axis is $x = 3$, which agrees with $-\frac{b}{2a}$.

Reading the three forms

Choosing the right form is a strategy:

For vertex form, watch the sign flip: $y = (x - 3)^2 + 4$ has $h = +3$, so the vertex is $(3, 4)$, not $(-3, 4)$. The constant inside the parentheses is subtracted, so its opposite is the vertex $x$.

Graphing a parabola point by point

To sketch a parabola, find the vertex first, then use symmetry to plot points on both sides. Because the parabola mirrors across its axis, a point at $x = h + 1$ has the same height as the point at $x = h - 1$. So once you have the vertex and one extra point, its mirror image comes for free.

For $y = x^2 - 4x + 3$: the axis is $x = 2$, the vertex is $(2, -1)$, and the y-intercept is $(0, 3)$. By symmetry, the point matching $(0, 3)$ across the axis $x = 2$ is $(4, 3)$. Plotting $(2, -1)$, $(0, 3)$, and $(4, 3)$, plus the zeros $(1, 0)$ and $(3, 0)$, gives an accurate sketch. On a technology-enhanced item you plot these key points directly on the grid.

Interpreting a parabola in context

A quadratic often models a projectile or an area, and the features carry meaning. For a height $h(t) = -16t^2 + 32t + 5$: the y-intercept $5$ is the initial height, the vertex gives the maximum height and the time it occurs, and the positive zero gives the time the object lands. The MCAS rewards naming which feature answers the question: "highest point" means the vertex, "when it hits the ground" means the positive zero, and "starting height" means the y-intercept.

Try this

Q1. Which way does $y = -3x^2 + 2x - 1$ open, and does it have a max or min?

• Cue. $a = -3 < 0$, so it opens down and has a maximum.

Q2. Find the axis of symmetry of $y = x^2 + 4x - 7$.

• Cue. $x = -\frac{4}{2} = -2$.