Lesson 1.2 · Unit 1 · F-IF.7
Nine graphs worth knowing before you need them
Every function studied this year is one of nine basic shapes, moved and stretched. Learning the nine now means that later units introduce no new graphing work at all, only new algebra. This lesson is a reference you will come back to.
- Linear, \( f(x) = x \). A straight line through the origin at \( 45^\circ \). Domain and range both all reals.
- Quadratic, \( f(x) = x^2 \). A parabola with vertex at the origin opening up. Domain all reals, range \( y \ge 0 \). Even.
- Cubic, \( f(x) = x^3 \). Through the origin, rising left to right with a flattening at zero. Domain and range all reals. Odd.
- Square root, \( f(x) = \sqrt{x} \). Half a sideways parabola starting at the origin. Domain \( x \ge 0 \), range \( y \ge 0 \).
- Cube root, \( f(x) = \sqrt[3]{x} \). Like the square root but continuing into the third quadrant. Domain and range all reals. Odd.
- Absolute value, \( f(x) = \left| x \right| \). A V with its corner at the origin. Domain all reals, range \( y \ge 0 \). Even.
- Reciprocal, \( f(x) = \dfrac{1}{x} \). Two branches with asymptotes on both axes. Domain and range all reals except zero. Odd.
- Exponential \( f(x) = 2^x \) and logarithmic \( f(x) = \log_2 x \), reflections of each other in the line \( y = x \), with a horizontal and a vertical asymptote respectively.
Where students lose marks: confusing the square root graph with half a parabola turned on its side and drawing it symmetric about the \( x \)-axis. \( y = \sqrt{x} \) has only the upper half, because the radical sign means the principal root. The full sideways parabola is \( x = y^2 \), which is not a function.
The problem. (a) State the domain and range of \( f(x) = \sqrt{x} \) and of \( g(x) = \sqrt[3]{x} \), and explain the difference. (b) Describe the end behavior of \( x^2 \), \( x^3 \) and \( \dfrac{1}{x} \). (c) Which of the nine are even, which odd, and which neither? (d) Explain why the exponential and logarithmic graphs are reflections in \( y = x \).
Step one: answer (a) for the square root. The radicand must be nonnegative, so the domain is \( x \ge 0 \). The radical sign denotes the principal, nonnegative root, so the range is \( y \ge 0 \). The graph occupies only the first quadrant and the origin.
Step two: answer (a) for the cube root. An odd root accepts negative inputs, because a negative number has a real cube root: \( \sqrt[3]{-8} = -2 \), since \( (-2)^3 = -8 \). So both the domain and the range are all real numbers.
Step three: explain the difference. Squaring destroys sign information: both \( 3 \) and \( -3 \) square to 9, so \( \sqrt{9} \) must choose one and chooses the positive. Cubing preserves sign, since a negative cubed stays negative, so the cube root can recover the input uniquely and needs no restriction. That single fact explains every later difference between even and odd roots, including the absolute value that appears when simplifying \( \sqrt{x^2} \).
Step four: end behavior of \( x^2 \) for (b). As \( x \to \infty \) the outputs grow without bound, and as \( x \to -\infty \) they also grow without bound, since squaring removes the sign. Both ends rise.
Step five: end behavior of \( x^3 \) and \( \dfrac{1}{x} \). For \( x^3 \), cubing preserves sign, so the outputs rise without bound to the right and fall without bound to the left. One end up, one end down. For \( \dfrac{1}{x} \), dividing 1 by an ever larger number gives an ever smaller result, so the outputs approach zero from above on the right and from below on the left. The \( x \)-axis is a horizontal asymptote at both ends, and the \( y \)-axis is a vertical asymptote at zero.
Step six: classify for (c). Even means \( f(-x) = f(x) \), a graph symmetric about the \( y \)-axis. Odd means \( f(-x) = -f(x) \), a graph unchanged by a \( 180^\circ \) rotation about the origin. Even: \( x^2 \) and \( \left| x \right| \). Odd: \( x \), \( x^3 \), \( \sqrt[3]{x} \) and \( \dfrac{1}{x} \). Neither: \( \sqrt{x} \), \( 2^x \) and \( \log_2 x \).
Step seven: check one of each. For \( \dfrac{1}{x} \): \( f(-x) = \dfrac{1}{-x} = -\dfrac{1}{x} = -f(x) \). Odd, confirmed. For \( \sqrt{x} \): \( f(-x) \) does not even exist for positive \( x \), so it cannot equal \( \pm f(x) \). Neither, and the domain alone settles it. Lesson 1.4 develops these tests properly.
Step eight: answer (d). The logarithm is defined as the inverse of the exponential: \( \log_2 y = x \) means exactly \( 2^x = y \). So the two functions swap the roles of input and output. A point \( (a, b) \) on the exponential graph means \( 2^a = b \), which means \( \log_2 b = a \), which is the point \( (b, a) \) on the logarithmic graph. Swapping coordinates is precisely reflection in the line \( y = x \), so the graphs are mirror images in that line. A concrete case: \( (3, 8) \) lies on \( y = 2^x \) since \( 2^3 = 8 \), and \( (8, 3) \) lies on \( y = \log_2 x \) since \( \log_2 8 = 3 \). Correct. That relationship also explains their asymptotes: the exponential's horizontal asymptote at \( y = 0 \) reflects into the logarithm's vertical asymptote at \( x = 0 \), which is why the logarithm's domain excludes zero and everything below it.
Problems 1 to 5 are routine. Problems 6 to 10 are multi-step or ask for reasoning.
- State the range of \( f(x) = x^2 \).
Show the full solution
\( y \ge 0 \)
- State the domain of \( f(x) = \sqrt[3]{x} \).
Show the full solution
Odd roots accept negatives. All reals
- State the range of \( f(x) = \left| x \right| \).
Show the full solution
\( y \ge 0 \)
- Name the asymptotes of \( f(x) = \dfrac{1}{x} \).
Show the full solution
\( x = 0 \) and \( y = 0 \)
- Which parent function has a corner rather than a smooth turn?
Show the full solution
Absolute value
- Compare the graphs of \( x^2 \) and \( x^3 \) for \( x \) between 0 and 1.
Show the full solution
On \( 0 \lt x \lt 1 \), raising to a higher power makes a number smaller, since multiplying by a fraction reduces it. At \( x = 0.5 \): \( x^2 = 0.25 \) and \( x^3 = 0.125 \). So the cubic lies below the quadratic on that interval, which reverses outside it: at \( x = 2 \), \( x^2 = 4 \) and \( x^3 = 8 \). They cross at \( x = 0 \) and \( x = 1 \), where both equal 0 and 1 respectively. The cubic is below the quadratic on \( (0, 1) \) and above it beyond 1
- State the domain and range of \( f(x) = 2^x \), and of \( f(x) = \log_2 x \).
Show the full solution
Exponential: domain all reals, range \( y \gt 0 \). No power of 2 is zero or negative, so the graph stays strictly above the \( x \)-axis. Logarithmic: domain \( x \gt 0 \), range all reals. Notice the swap: the exponential's domain is the logarithm's range and the exponential's range is the logarithm's domain. That is what being inverses means. Exponential: all reals, \( y \gt 0 \). Logarithmic: \( x \gt 0 \), all reals
- Sketch-describe \( y = \sqrt{x} \) and \( y = x^2 \) on the same axes for \( x \ge 0 \),
and say where they cross.
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Both start at the origin and rise. The square root rises quickly at first and then flattens; the parabola starts flat and then rises steeply. They cross where \( \sqrt{x} = x^2 \). Squaring both sides gives \( x = x^4 \), so \( x^4 - x = 0 \) and \( x(x^3 - 1) = 0 \), giving \( x = 0 \) or \( x = 1 \). Check both in the original: \( \sqrt{0} = 0 = 0^2 \) and \( \sqrt{1} = 1 = 1^2 \). Both genuine, neither extraneous. Between 0 and 1 the square root is above; beyond 1 the parabola is above. At \( x = 0.25 \): \( \sqrt{x} = 0.5 \) and \( x^2 = 0.0625 \). Confirmed. They cross at \( (0,0) \) and \( (1,1) \)
- Explain why knowing these nine graphs makes the rest of the year easier.
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Every function in Algebra 2 is one of these nine with some combination of shifts, reflections and stretches applied, and lesson 1.3 shows that those transformations change a graph in entirely predictable ways. So graphing a new function is never a fresh problem. It is two questions: which parent is this, and what has been done to it. A student who can answer both can sketch \( y = -3\sqrt{x + 4} - 1 \) without plotting a single point. What that buys in each later unit. Unit 2's parabolas are the quadratic parent transformed. Unit 4's rational functions start from the reciprocal parent. Unit 5's radical functions are the square root and cube root parents. Units 6 and 7 are the exponential and logarithmic parents. Unit 9's sinusoids add one more parent and reuse the same four transformations. The deeper point. Organizing the course this way is a claim about mathematics, not a study tip: the transformations form a structure that acts the same way on every function, so learning it once covers every family. That is why this unit comes first rather than being tucked into whichever chapter happens to need it. Every later function is one of these nine transformed, so graphing reduces to identifying the parent and the transformation
- For each of \( x^2 \), \( x^3 \), \( \sqrt{x} \) and \( \dfrac{1}{x} \), state whether it
has an inverse function on its natural domain, and if not, what restriction fixes it.
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The test. A function has an inverse function exactly when no output is repeated, which on a graph is the horizontal line test: no horizontal line may meet the graph twice. \( x^2 \). Fails. The line \( y = 4 \) meets it at \( x = 2 \) and \( x = -2 \). Restricting the domain to \( x \ge 0 \) removes the left branch and makes it one-to-one; the inverse is then \( \sqrt{x} \). Restricting to \( x \le 0 \) works equally well and gives \( -\sqrt{x} \). \( x^3 \). Passes. The cubic is increasing everywhere, so no output repeats, and its inverse is \( \sqrt[3]{x} \) on all reals with no restriction needed. \( \sqrt{x} \). Passes on its natural domain \( x \ge 0 \), since it is increasing. Its inverse is \( x^2 \) restricted to \( x \ge 0 \), and that restriction on the inverse is the mirror image of the restriction that was needed on \( x^2 \) in the first place. \( \dfrac{1}{x} \). Passes. Although it has two branches, no output is repeated: the right branch produces only positive values and the left only negative. Its inverse is itself, since solving \( y = \dfrac{1}{x} \) for \( x \) gives \( x = \dfrac{1}{y} \). A function equal to its own inverse is called an involution, and its graph is symmetric about the line \( y = x \). The pattern. The even-powered and even-root functions need restrictions; the odd-powered and odd-root ones do not. It is the same sign-destroying property that separated \( \sqrt{x} \) from \( \sqrt[3]{x} \) in the worked example. \( x^3 \), \( \sqrt{x} \) and \( \frac{1}{x} \) have inverses as they stand; \( x^2 \) needs its domain restricted to \( x \ge 0 \)