10.1: Intercepts and Base Graphs
- Page ID
- 174227
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Definitions and Theorems
Every graph in this text is read against two reference lines: the horizontal \(x\)-axis and the vertical \(y\)-axis. The points at which a graph meets these axes are its primary landmarks.
The \(y\)-intercept of the graph of a function or relation is a point at which the graph intersects the \(y\)-axis. Every such point satisfies \(x=0\). For a function \(f\), the \(y\)-intercept, if it exists, is the point \((0,f(0))\).
An \(x\)-intercept of the graph of a function or relation is a point at which the graph intersects the \(x\)-axis. Every such point satisfies \(y=0\), and so has the form \((a,0)\) where \(f(a)=0\). The values \(a\) are called the zeros (or roots) of \(f\).
Let \(f\) be a function.
- The graph of \(f\) has at most one \(y\)-intercept, obtained by evaluating \(f\) at \(0\); that is, the \(y\)-intercept is \((0,f(0))\) whenever \(0\) is in the domain of \(f\).
- The \(x\)-intercepts of the graph of \(f\) are the points \((a,0)\) for which \(f(a)=0\). A function may have no \(x\)-intercept, exactly one, or several.
Only the real solutions of \(f(x)=0\) correspond to \(x\)-intercepts, because only real numbers locate points on the \(x\)-axis. A function such as \(f(x)=x^2+4\) has zeros, but they are not real; its graph has no \(x\)-intercept.
The Library of Basic Functions
A returning student is expected to recognize a small collection of base graphs on sight. Each belongs to a family of related graphs generated by shifting, reflecting, or stretching a single simplest member.
A parent function is the simplest representative of a family of functions, expressed without any translation, reflection, or dilation. Every other member of the family is obtained from the parent function by such transformations.
The four parent functions below occur throughout the calculus sequence. Commit their shapes, domains, ranges, and basic points to memory.
| Parent Function | Domain | Range | Shape and Basic Points |
|---|---|---|---|
| Squaring: \(f(x)=x^2\) | \((-\infty,\infty)\) | \([0,\infty)\) | Upward parabola, vertex \((0,0)\); passes through \((\pm 1,1)\) and \((\pm 2,4)\). |
| Cubing: \(f(x)=x^3\) | \((-\infty,\infty)\) | \((-\infty,\infty)\) | Symmetric about the origin; passes through \((\pm 1,\pm 1)\) and \((\pm 2,\pm 8)\). |
| Absolute value: \(f(x)=|x|\) | \((-\infty,\infty)\) | \([0,\infty)\) | V-shape, vertex \((0,0)\); passes through \((\pm 1,1)\) and \((\pm 2,2)\). |
| Square root: \(f(x)=\sqrt{x}\) | \([0,\infty)\) | \([0,\infty)\) | Half-parabola from \((0,0)\); passes through \((1,1)\), \((4,2)\), and \((9,3)\). |
The value \(c\) in the domain of the function \(f\) is called a zero of \(f\) if \(f(c) = 0\).
If \(c\) solves a given equation, we say that \(c\) is a root of the equation.
Examples
Find the \(y\)-intercept of \(f(x)=2x^3-4x+7\).
- Solution
- The \(y\)-intercept is found by evaluating the function at \(x=0\):\[f(0)=2(0)^3-4(0)+7=7.\nonumber\]The \(y\)-intercept is the point \((0,7)\).
Find the \(x\)-intercepts of \(f(x)=x^3-4x\).
- Solution
- Set the function equal to \(0\) and solve, since the \(x\)-intercepts occur where \(y=0\):\[x^3-4x=0.\nonumber\]Factor out the common factor \(x\), then factor the resulting difference of squares:\[x(x^2-4)=x(x-2)(x+2)=0.\nonumber\]By the zero-product property, \(x=0\), \(x=2\), or \(x=-2\). The graph therefore has three \(x\)-intercepts: \((-2,0)\), \((0,0)\), and \((2,0)\).
Find every intercept of \(f(x)=x^2-5x+6\).
- Solutions
-
\(y\)-intercept. Evaluate at \(x=0\):\[f(0)=(0)^2-5(0)+6=6,\nonumber\]giving the point \((0,6)\).
\(x\)-intercepts. Set \(f(x)=0\) and factor:\[x^2-5x+6=(x-2)(x-3)=0.\nonumber\]Hence \(x=2\) or \(x=3\), giving the points \((2,0)\) and \((3,0)\).
Find every intercept of \(f(x)=x^2+4\).
- Solutions
-
\(y\)-intercept. Evaluate at \(x=0\):\[f(0)=(0)^2+4=4,\nonumber\]giving the point \((0,4)\).
\(x\)-intercepts. Set \(f(x)=0\):\[x^2+4=0 \quad\Longrightarrow\quad x^2=-4.\nonumber\]No real number has a square of \(-4\), so this equation has no real solution. The graph has no \(x\)-intercept.
Find every intercept of \(f(x)=\sqrt{x}-2\).
- Solutions
-
The domain of \(f\) is \([0,\infty)\), since the radicand must be nonnegative.
\(y\)-intercept. Evaluate at \(x=0\):\[f(0)=\sqrt{0}-2=-2,\nonumber\]giving the point \((0,-2)\).
\(x\)-intercept. Set \(f(x)=0\) and isolate the radical:\[\sqrt{x}-2=0 \quad\Longrightarrow\quad \sqrt{x}=2.\nonumber\]Squaring both sides gives \(x=4\), which lies in the domain, so the \(x\)-intercept is \((4,0)\).
State the parent function of each of the following.
- \(f(x)=\sqrt{x+5}\)
- \(f(x)=-3|x|\)
- \(f(x)=(x-1)^3\)
- Solutions
-
- Removing the horizontal shift of \(5\) units leaves the square root parent function, \(f(x)=\sqrt{x}\).
- Removing the reflection and vertical stretch by \(-3\) leaves the absolute value parent function, \(f(x)=|x|\).
- Removing the horizontal shift of \(1\) unit leaves the cubing parent function, \(f(x)=x^3\).
Sources
Several parts of this text use modifications from the following sources:
- Wikipedia article: "Y-intercept"
- Wikipedia article: "Parent function"
All of these sources are released under the Creative Commons Attribution-Share-Alike License 4.0.