Shapes of Graphs Flashcards

All 18 cards in this deck

What are the coordinates of the point where the curve $y = ax^2 + bx + c$ crosses the $y$-axis?

e.g. $y = x^2 + 2x - 5$

$(0, c)$ — substitute $x = 0$.

e.g. $(0, -5)$

What are the coordinates of the turning point of $y = a(x - p)^2 + q$?

e.g. $y = 4 - (x - 3)^2$

$(p, q)$

e.g. maximum point $(3, 4)$

What is the equation of the line of symmetry of $y = a(x - p)^2 + q$?

e.g. $y = 2(x + 1)^2 - 7$

$x = p$

e.g. $x = -1$

True or false? The graph of $y = 2x^2 - 8x + 1$ has a maximum point.

False. The coefficient of $x^2$ is positive, so the curve is U-shaped and has a minimum point.

What are the steps to sketch a quadratic, labelling its intersections with the axes?

e.g. $y = x^2 - 4x + 3$

1. Shape from the $x^2$ coefficient: positive, so U-shaped.
2. $x = 0$ for the $y$-intercept: $(0, 3)$.
3. $y = 0$ for the roots: $(1, 0)$ and $(3, 0)$.

What does the graph of $y = ab^x$ (with $a > 0$ and $b > 1$) look like?

An increasing curve through $(0, a)$, rising steeply to the right and approaching the $x$-axis on the left.

e.g. $y = 3 \times 2^x$ passes through $(0, 3)$

How is the graph of $y = ab^{-x}$ related to the graph of $y = ab^{x}$?

It is its reflection in the $y$-axis, since $ab^{-x} = a\left(\frac{1}{b}\right)^x$.

e.g. $y = 3 \times 2^{-x}$ is $y = 3 \times 2^{x}$ reflected in the $y$-axis

True or false? The graph of $y = 3 \times 2^{-x}$ falls as $x$ increases and eventually crosses the $x$-axis.

False. It gets closer and closer to the $x$-axis but never touches or crosses it.

What are the steps to find $a$ and $b$ for a curve $y = ab^x$ passing through two given points?

e.g. the curve passes through $(0, 5)$ and $(2, 45)$

1. Use the point with $x = 0$ to get $a$: $a = 5$.
2. Substitute the other point: $5b^2 = 45$, so $b^2 = 9$.
3. Take the positive root: $b = 3$, so $y = 5 \times 3^x$.

What are the steps to find the $x$-coordinate of a point on $y = ab^x$ with a known $y$-value?

e.g. find $x$ when $y = 96$ on $y = 3 \times 2^x$

1. Divide by $a$ to isolate the power: $2^x = 32$.
2. Write the number as a power of $b$: $32 = 2^5$.
3. Equate the exponents: $x = 5$.

What are the steps to find the straight line to draw on a given curve to solve a different equation?

e.g. graph of $y = x^2 - 3x + 1$ is drawn; solve $x^2 - 4x = 0$

1. Write the equation to be solved with $0$ on one side: $x^2 - 4x = 0$.
2. Subtract it from the curve's equation: $(x^2 - 3x + 1) - (x^2 - 4x) = x + 1$.
3. Draw $y = x + 1$ and read off the $x$-coordinates of the intersections.

When a straight line is drawn on a curve to solve an equation, which coordinates of the intersection points are the solutions?

The $x$-coordinates only; the $y$-values are ignored.

True or false? If the line needed to solve an equation is $y = 3 - \frac{x}{2}$, drawing $y = \frac{x}{2} - 3$ instead would still be correct.

False. The reversed/negative form is a different line; only $y = 3 - \frac{x}{2}$ (or a rearrangement of it such as $2y = 6 - x$) is accepted.

True or false? At a maximum point on a curve the gradient is zero, so the tangent there is parallel to the $x$-axis.

True. The curve is momentarily flat at a maximum point.

True or false? A curve that touches the $x$-axis at its minimum point $(a, 0)$ and does not meet the $x$-axis anywhere else has three different values of $x$ for which $y = 0$.

False. Touching the axis there gives only one value of $x$ with $y = 0$, namely $x = a$.

What does it mean to say a function is increasing over a section of its graph?

e.g. between a minimum point and the next maximum point

$y$ gets larger as $x$ gets larger — the curve goes upwards from left to right.

e.g. between a minimum and the following maximum the function is increasing

What is a piecewise-defined function?

A function given by a different rule on each of two or more separate domains (intervals) of $x$.
e.g. $y = 2x$ for $0 \le x \le 2$ and $y = 4$ for $2 < x \le 5$

What are the steps to draw the graph of a piecewise-defined function?
e.g. $y = 2x$ for $0 \le x \le 2$, then $y = 4$ for $2 < x \le 5$

1. Work out the end values of each piece: $(0,0)$ and $(2,4)$; then $(2,4)$ and $(5,4)$.
2. Draw each rule only across its own domain: line from $(0,0)$ to $(2,4)$, then a horizontal line to $(5,4)$.
3. Draw nothing outside the given domains.