Sketch the graph of each function algebra 1
Standard Algebra 1 graph sketches emphasize the function family (line, parabola, V-shape, radical curve, exponential curve) and a small set of reliable features: intercepts, turning points, symmetry, and a few computed points.
Representative function set and key features
| Function | Family and defining feature | Anchor points (for a quick sketch) |
|---|---|---|
| \(f(x)=2x-3\) | Linear; constant slope \(2\) and y-intercept \(-3\) | \((0,-3)\), \((2,1)\), x-intercept \((1.5,0)\) |
| \(g(x)=-(x-1)^2+4\) | Quadratic; vertex form with vertex \((1,4)\), opens downward | Vertex \((1,4)\), x-intercepts \((-1,0)\), \((3,0)\), y-intercept \((0,3)\) |
| \(h(x)=|x+2|-1\) | Absolute value; V-shape with vertex \((-2,-1)\) | Vertex \((-2,-1)\), x-intercepts \((-3,0)\), \((-1,0)\), y-intercept \((0,1)\) |
| \(p(x)=\sqrt{x+1}-2\) | Square root; starts at the left endpoint \((-1,-2)\) | \((-1,-2)\), \((0,-1)\), \((3,0)\), \((8,1)\) |
| \(q(x)=2^{\,x-1}\) | Exponential; horizontal asymptote \(y=0\), always positive | \((0,0.5)\), \((1,1)\), \((2,2)\), \((3,4)\) |
Graph features that determine the sketch
- Intercept structure
- x-intercepts satisfy \(y=0\); y-intercepts satisfy \(x=0\). Linear and quadratic graphs often become accurate with only intercepts and one additional point.
- Turning points and symmetry
- Quadratics have a vertex and an axis of symmetry. Absolute value graphs have a vertex and two rays with equal steepness on each side.
- Domain and range restrictions
- Radical graphs have a left endpoint or boundary from the square-root requirement. Exponential graphs have positive outputs and a horizontal asymptote at \(y=0\).
Answer key (rigorous details)
\(f(x)=2x-3\) (linear)
Slope \(m=2\) and y-intercept \(-3\) place the line through \((0,-3)\) with a rise of \(2\) for every run of \(1\).
x-intercept: \[ 0 = 2x - 3 \quad \Rightarrow \quad x = \frac{3}{2}. \] Domain: all real \(x\). Range: all real \(y\).
\(g(x)=-(x-1)^2+4\) (quadratic)
Vertex form gives vertex \((1,4)\) and axis of symmetry \(x=1\). The negative leading sign indicates a downward-opening parabola.
x-intercepts: \[ 0 = -(x-1)^2 + 4 \;\Rightarrow\; (x-1)^2 = 4 \;\Rightarrow\; x = -1,\; 3. \] y-intercept: \(g(0)=-(1)^2+4=3\). Domain: all real \(x\). Range: \(y \le 4\).
\(h(x)=|x+2|-1\) (absolute value)
The expression \(|x+2|\) shifts the basic V-shape left by \(2\). Subtracting \(1\) shifts downward by \(1\). Vertex: \((-2,-1)\).
x-intercepts: \[ 0 = |x+2| - 1 \;\Rightarrow\; |x+2| = 1 \;\Rightarrow\; x=-3,\; -1. \] Domain: all real \(x\). Range: \(y \ge -1\).
\(p(x)=\sqrt{x+1}-2\) (square root)
The inside shift \(x+1\) moves the basic square-root curve left by \(1\). The \(-2\) shifts the graph downward by \(2\). Left endpoint: \((-1,-2)\).
Domain: \[ x+1 \ge 0 \;\Rightarrow\; x \ge -1. \] x-intercept: \[ 0=\sqrt{x+1}-2 \;\Rightarrow\; \sqrt{x+1}=2 \;\Rightarrow\; x=3. \] Range: \(y \ge -2\).
\(q(x)=2^{\,x-1}\) (exponential)
The shift \(x-1\) moves the base graph \(2^x\) right by \(1\). Outputs remain positive, and the curve approaches \(y=0\) as \(x\) decreases.
y-intercept: \(q(0)=2^{-1}=0.5\). Standard points: \(q(1)=1\), \(q(2)=2\), \(q(3)=4\). Domain: all real \(x\). Range: \(y>0\). Horizontal asymptote: \(y=0\).
Common errors (Algebra 1 graph sketches)
- Scale mismatch: unequal unit spacing between the x-axis and y-axis causing distorted slopes and vertex locations.
- Intercept omission: missing x-intercepts or y-intercepts when they exist and are easy to compute.
- Vertex displacement: incorrect sign on \((x-h)\) in vertex form or inside an absolute value or radical.
- Domain restriction loss: drawing square-root graphs for \(x<-1\) in \(p(x)=\sqrt{x+1}-2\) or allowing negative outputs for \(q(x)=2^{\,x-1}\).