AP Precalculus Archives - Good Grade This Way

Quadratic Functions A quadratic function has the form $latex f(x)=ax^2+bx+c$ with $latex a\neq 0$. Its graph is a parabola whose shape and position are controlled by $latex a,\,b,\,c$. Three Forms at a Glance Standard form: $latex f(x)=ax^2+bx+c$. Vertex form: $latex f(x)=a(x-h)^2+k$ (vertex $latex (h,k)$, axis $latex x=h$). Factored form: $latex f(x)=a(x-p)(x-q)$ (zeros $latex x=p,q$, axis $latex x=\displaystyle\frac{p+q}{2}$). Core Features Opens: up if $latex a>0$, down if $latex a1$, wider if $latex |a|0$ (opens up): vertex value $latex f\!\left(-\displaystyle\frac{b}{2a}\right)$ is a minimum; domain $latex (-\infty,\infty)$; range $latex \left[y_{\min},\infty\right)$ with $latex y_{\min}=f\!\left(-\displaystyle\frac{b}{2a}\right)$; decreasing left of the vertex, increasing right. If $latex a0$: two real, distinct zeros → two $latex x$-intercepts. $latex D=0$: one real, repeated zero (touches the axis at the vertex). $latex D<0$: no real zeros (parabola lies entirely above or below the $latex x$-axis). Vieta’s Relations For real/complex roots $latex r_1,\,r_2$, $latex r_1+r_2=-\displaystyle\frac{b}{a}$ and $latex r_1r_2=\displaystyle\frac{c}{a}$. Standard vs. Vertex vs. Factored FormEquationAxis of SymmetryVertex$x$-Intercepts$y$-InterceptBest ForStandard$latex f(x)=ax^2+bx+c$$latex x=-\displaystyle\frac{b}{2a}$$latex \left(-\displaystyle\frac{b}{2a},\,f\!\left(-\displaystyle\frac{b}{2a}\right)\right)$Use quadratic formula / factoring if possible$latex (0,c)$Reading coefficients; applying formulaeVertex$latex f(x)=a(x-h)^2+k$$latex x=h$$latex (h,k)$Solve $latex a(x-h)^2+k=0$$latex (0,\,a h^2+k)$Graphing min/max, range, transformationsFactored$latex f(x)=a(x-p)(x-q)$$latex x=\displaystyle\frac{p+q}{2}$$latex \left(\displaystyle\frac{p+q}{2},\,f\!\left(\displaystyle\frac{p+q}{2}\right)\right)$$latex (p,0)$ and $latex (q,0)$$latex \big(0,\,a\,pq\big)$Finding zeros and axis quickly Quick Graphing Checklist Identify $latex a$ (opening & width): up/down, narrower/wider. Find axis $latex x=-\displaystyle\frac{b}{2a}$ (or $latex x=h$ in vertex form). Compute vertex value $latex f\!\left(-\displaystyle\frac{b}{2a}\right)$ (or read $latex k$). Get intercepts: $latex y$-intercept $latex (0,c)$; $latex x$-intercepts by solving $latex f(x)=0$. Plot symmetric points about the axis and sketch the parabola. Common Pitfalls Forgetting the negative in $latex x=-\displaystyle\frac{b}{2a}$. Completing the square without factoring out $latex a$ first when $latex a\neq 1$. Assuming intercepts exist even when $latex D<0$. Practice Given $latex f(x)=2x^2-8x+5$, find the axis, vertex, and range. Verify whether it opens up or down and whether it is narrower or wider than $latex y=x^2$. Convert $latex f(x)=-3x^2+12x-7$ to vertex form by completing the square. State $latex (h,k)$ and the max/min value. A quadratic has zeros $latex x=1$ and $latex x=7$ and passes through $latex (0,14)$. Find $latex a$ and write $latex f(x)$ in factored and standard forms. For $latex f(x)=x^2+4x+c$, find all values of $latex c$ for which the graph touches the $latex x$-axis (one real solution). Then determine $latex c$ for which $latex f(x)=0$ has no real solution. Solve $latex 5x^2-2x-3=0$ and interpret your solutions as $latex x$-intercepts of the parabola. State the discriminant and what it implies.

Tag: AP Precalculus

The Ultimate Guide to Quadratic Functions and Equations (Precalculus Made Easy)

AP Precalculus – Inverse Functions and Everything You Need to Know

Do I need AP Precalculus?