Graphs of Functions Flashcards

All 21 cards in this deck

How do you tell from its equation whether a graph is linear, quadratic, cubic or reciprocal?

e.g. $y = x^3 - 2$

Look at the highest power of $x$ (or $\frac{1}{x}$ for reciprocal).

$y = x^3 - 2$ has highest power $3$, so it is a cubic.

What does the graph of the reciprocal function $y = \frac{1}{x}$ look like?

Two separate curves in opposite quadrants, getting closer and closer to the axes but never touching them.

What does the graph of $y = x^3$ look like?

A curve rising from bottom left to top right, passing through the origin, flattening out as it crosses there.

True or false? The graph of $y = -x^2$ is a U-shaped curve.

False. A negative $x^2$ term gives an upside-down (n-shaped) parabola.

True or false? The graph of $y = 5 - 2x$ is a straight line.

True. The highest power of $x$ is $1$, so the graph is linear.

On the graph of a quadratic, where do you read off the roots?

The $x$-values where the curve crosses the $x$-axis.

What is the turning point of a quadratic graph?

The point where the curve changes direction — the minimum point of a U-shape or the maximum point of an n-shape. Written as coordinates, e.g. $(1, -4)$.

What is the $y$-intercept of the graph of $y = x^2 + bx + c$?

e.g. $y = x^2 - 6x + 4$

The constant term $c$ (the value of $y$ when $x = 0$).

For $y = x^2 - 6x + 4$ the $y$-intercept is $4$.

What are the steps to use a quadratic graph to find the values of $x$ for a given $y$?

e.g. find $x$ when $y = 3$

1. Draw a horizontal line at that $y$-value: the line $y = 3$.
2. Mark where this line meets the curve (usually two points).
3. Read straight down to the $x$-axis for each $x$-value.

How do you use a quadratic graph to find the value of $y$ for a given $x$?

e.g. find $y$ when $x = 1.5$

Read up from $x = 1.5$ on the $x$-axis to the curve, then across to the $y$-axis and read off $y$.

True or false? Every quadratic graph crosses the $x$-axis twice, so it always has two roots.

False. A quadratic graph can cross the $x$-axis twice, touch it once, or not reach it at all (no roots).

What are the steps to find a missing $y$-value in a table of values for a quadratic?

e.g. $y = x^2 - x - 2$ when $x = -2$

1. Substitute the $x$-value: $(-2)^2 - (-2) - 2$.
2. Work out the power first: $4 + 2 - 2$.
3. Add and subtract: $y = 4$.

True or false? When $x = -3$, the value of $x^2$ is $-9$.

False. $(-3)^2 = 9$ — squaring a negative number gives a positive result.

What are the steps to draw a graph from a completed table of values?

e.g. $y = x^2 - x - 2$ for $x$ from $-2$ to $3$

1. Plot each $(x, y)$ pair from the table as a point.
2. Join them with a single smooth freehand curve, not straight line segments.
3. Keep the curve within the given range, $x = -2$ to $x = 3$.

A student draws the graph of $y = x^2 + 1$ by joining her plotted points with a ruler. What is wrong with her graph?

It is made of straight line segments — it should be a smooth curve drawn freehand.

How do you complete a table of values for a reciprocal function $y = \frac{k}{x}$?

e.g. $y = \frac{6}{x}$ when $x = 0.5$

Divide $k$ by each $x$-value.

$6 \div 0.5 = 12$, so $y = 12$.

Which value of $x$ can never appear in a table of values for $y = \frac{k}{x}$?

$x = 0$ — you cannot divide by zero, so the graph has no point there.

When drawing a graph from a completed table of values, what does each column of the table give you?

e.g. the column $x = -2$, $y = 4$

One coordinate pair to plot.

e.g. plot the point $(-2, 4)$.

Two neighbouring points in a quadratic table have the same lowest $y$-value. Where should the curve go between them?

e.g. $y = -2$ at both $x = 0$ and $x = 1$

It should dip below them, reaching its minimum halfway between the two $x$-values.

e.g. the lowest point is below $y = -2$, at $x = 0.5$.

True or false? A curve drawn from a table of values is acceptable as long as it passes close to the plotted points.

False. The curve must pass through every plotted point.

What is wrong with a quadratic graph drawn with a sharp point at the bottom?

A quadratic curve must be smooth and rounded at the turning point, not pointed.