The vertex form of a quadratic function is:
\$\$f(x) = a(x-h)^2 + k\$\$
where:
- \$a\$ is the leading coefficient.
- \$(h, k)\$ is the vertex of the parabola.
The vertex form of a quadratic function is useful because it directly provides the vertex of the parabola, which is the point where the parabola changes direction.
NoteThe vertexof a parabola is the point \$(h, k)\$ where the parabola changes direction. It is the lowest point on the graph if the parabola opens upwards, and the highest point if it opens downwards.
Graphing Quadratic Functions in Vertex Form
To graph a quadratic function in vertex form:
- Identify the vertex \$(h, k)\$.
- Determine the direction of the parabola:
- If \$a > 0\$, the parabola opens upwards.
- If \$a < 0\$, the parabola opens downwards.
- Plot the vertex on the coordinate plane.
- Use the leading coefficient \$a\$ to determine the width of the parabola:
- If \$|a| > 1\$, the parabola is narrower.
- If \$0 < |a| < 1\$, the parabola is wider.
- Plot additional points by choosing \$x\$-values and calculating the corresponding \$y\$-values.
- Reflect the points across the axis of symmetry (the vertical line \$x = h\$).
- Draw a smooth curve through the points to complete the parabola.
The axis of symmetryis the vertical line \$x = h\$ that passes through the vertex. It divides the parabola into two symmetric halves.
Finding Intercepts
\$x\$-Intercepts
To find the \$x\$-intercepts of a quadratic function in vertex form, set \$f(x) = 0\$ and solve for \$x\$:
\$\$a(x-h)^2 + k = 0\$\$
- Subtract \$k\$ from both sides: \$a(x-h)^2 = -k\$.
- Divide by \$a\$: \$(x-h)^2 = -\frac{k}{a}\$.
- Take the square root of both sides: \$x-h = \pm \sqrt{-\frac{k}{a}}\$.
- Solve for \$x\$: \$x = h \pm \sqrt{-\frac{k}{a}}\$.
The \$x\$-intercepts are the points where the parabola crosses the \$x\$-axis. If \$-k/a < 0\$, the quadratic function has no real \$x\$-intercepts.
\$y\$-Intercept
To find the \$y\$-intercept, set \$x = 0\$ and solve for \$f(0)\$:
\$\$f(0) = a(0-h)^2 + k = ah^2 + k\$\$
Self review1. Graph the quadratic function \$f(x) = -\frac{1}{2}(x-2)^2 + 4\$. 2. Find the \$x\$-intercepts and \$y\$-intercept of the function. 3. Verify your graph by checking the symmetry and intercepts.
Theory of KnowledgeHow does the vertex form of a quadratic function help us understand the transformation of the graph? What other forms of quadratic functions can be used to analyze their properties?
